I am working on a campaign that deals with crime bosses and prohibition era mob families and I want the players session zero to be them being sent to do a job for their boss. The twist here is that either the job was botched or the boss was secretly sending them on a mission intending for them to be killed. This would then be a good setup for them going forward through the campaign, at least for the first arc, either they could try to regain favor with the boss, or maybe find out the truth and try and take him down.

So how exactly would I design a session/encounter in a way that when they "fail" it the players do not feel as though they were being railroaded?

You tell them you're doing it, before the game starts.

There's no secret to railroading the players without them being railroaded.  Either your players are okay with a predetermined set-piece, or they're not.  I'd ask, in the actual Session Zero: "Hey - do you guys want to play a Doomed Mission?  The story picks up with you guys in disgrace after a failed mission, but I think it'd be cool to run that mission and let you guys try it, like the Arasaka heist in Cyberpunk 2077.  Would you be interested in that, or would you prefer to start after the failed mission so you don't have to lose?"

That makes sense. But wouldn't that kind of ruin the potential for having the players feel driven throughout the campaign? Or even ruin the surprise of finding out later on that it was a setup from the beginning?

You don't have to tell the players why they failed.  Simply tell them that your story begins with them as a band of disgraced criminals after a failed job, then ask them if they'd like to play that doomed-to-failure job.  If they would?  Then they play the job, and you make sure it fails however you have to.  Why it fails, whether the players were set up or just bad at their job, or whether it was both?  That's for you the DM to decide, assuming the players are as terrible at crime as most D&D players are.  All the players will know is "we were given a job and it went south, and now we're in disgrace and have to claw our way out of the bottom of the pit again."

Quote from Darastrix >>

So how exactly would I design a session/encounter in a way that when they "fail" it the players do not feel as though they were being railroaded?

They are being railroaded. As Yurei1453 said, just own it. There's no point in trying to pretend they're not being railroaded, especially when the railroading consists of getting caught in a setup and failing. You don't know what they're going to do if they're not aware they're doomed to fail, and if they choose to do something that forces you to Deus Ex Machina the whole thing so they and up where you intended them to end up (rather than, say, all on a slab in the morgue) it's never going to feel right. 

For me the real question is whether it's worth it. If you tell them, out-of-game info, that their characters will end up captured or fleeing for their lives or whatever you have in mind, they're going to wonder why. They'll probably get curious about what the point of it is. If you instead just start in media res, have a regular session zero and at the start of session one tell them what the situation is, then they'll likely have some different ideas about what's going on. What I'd try to figure out is which of the two is better for the campaign. It'll probably depend a lot on how much suspicions you want them to have about the betrayal, and how you're going to convey that in either scenario. 

Quote from Yurei1453 >>

You don't have to tell the players why they failed.  Simply tell them that your story begins with them as a band of disgraced criminals after a failed job, then ask them if they'd like to play that doomed-to-failure job.  If they would?  Then they play the job, and you make sure it fails however you have to.  Why it fails, whether the players were set up or just bad at their job, or whether it was both?  That's for you the DM to decide, assuming the players are as terrible at crime as most D&D players are.  All the players will know is "we were given a job and it went south, and now we're in disgrace and have to claw our way out of the bottom of the pit again."

Quote from pangurjan >>

Quote from Darastrix >>

So how exactly would I design a session/encounter in a way that when they "fail" it the players do not feel as though they were being railroaded?

They are being railroaded. As Yurei1453 said, just own it. There's no point in trying to pretend they're not being railroaded, especially when the railroading consists of getting caught in a setup and failing. You don't know what they're going to do if they're not aware they're doomed to fail, and if they choose to do something that forces you to Deus Ex Machina the whole thing so they and up where you intended them to end up (rather than, say, all on a slab in the morgue) it's never going to feel right. 

For me the real question is whether it's worth it. If you tell them, out-of-game info, that their characters will end up captured or fleeing for their lives or whatever you have in mind, they're going to wonder why. They'll probably get curious about what the point of it is. If you instead just start in media res, have a regular session zero and at the start of session one tell them what the situation is, then they'll likely have some different ideas about what's going on. What I'd try to figure out is which of the two is better for the campaign. It'll probably depend a lot on how much suspicions you want them to have about the betrayal, and how you're going to convey that in either scenario. 

 You both make sound arguments. I will take this into consideration as I begin planning this out.

Thank you.

Quote from Darastrix >>

But wouldn't that kind of ruin the potential for having the players feel driven throughout the campaign? Or even ruin the surprise of finding out later on that it was a setup from the beginning?

As the others already touched upon, the players don't necessarily need to know the how and why their characters are going to fail this initial session/encounter. Heck, part of the fun for the players (and even yourself) might be to play through the session to actually determine the how and why they ended up failing.

Dice are typically random, and players frequently surprises their DM without their chosen actions. Despite knowing they'll ultimately have to fail, they provide you more details and background that feed into the bigger story with how they handle the session... Plus, while the characters might be doomed to fail, the players might still have some control over how badly they fail at this mission. Players might enjoy trying to either mitigate the amount of damage done by their doomed mission or they might want to fail so epically bad that it'll create some stories of its own.

Ask yourself what the players might succeed at in the "failed mission". If you can't come up with anything the players can change in the story or "succeed" at, I'm not sure it's worth playing it out. If they are sent on a mission to be "killed", the crime boss' plan obviously failed while the PCs succeeded. Focus on that, on how they survived this "doomed" mission. That can be fun to play out. The set up before that I would have done as quick cut scenes or something. 

Not to put too fine a point on what others have said, but I'd start them just after the failed mission. Then it weirdly doesn't feel like a railroad. Kind of how if you start with everyone in jail, they just go with it (as long as you promise them they'll find their gear soon enough), but if you start with them out of jail and being arrested, players will flip out at the railroad and being put in an unwinnable situation. My philosophy is that DMs are allowed quite a bit of leeway in railroading pre-session 1, as they work to get the party in the same place at the same time and set off the complication that will start to drive the story.  But once the session starts, then it's more up to the players to decide what they want to do with the situation the DM has placed them in to start. So if you tell them they're starting just after a mission went sideways, they say OK. But if you put them in a mission with no hope of success, they'll get upset -- rightly so, imo (in general, there will, of course, always be exceptions).

Also, not to get pedantic, what you're describing isn't really a session 0, its a session 1. A session 0 actually has little to no playing, and instead is a chance to discuss metagame issues, house rules and table rules and what kind of game everyone wants to play. During that session 0, is where you say something like, I'm thinking about starting the campaign just after your gang failed a big mission, cool?

So I think you're treating what you think of as a Session 0 as what's actually narratively your game's "prologue."  In actual Session 0 this would a be a back story you and the players agree upon to get the actual plot rolling.  Maybe investigating why things went bad, discovering they were set up etc.

As you seem to feel and you're definitely feeling from the thread so far, a lot of D&D players see "railroading" purely pejoratively.  I don't think that's a correct interpretation because if you reduce the "DM can't tell me what I'm doing" to the absurd the game never starts due to some sort of narrative Xeno's paradox.  

Besides, what's the difference between "railroading" and media res?  Media res is a convention that literally means "in the middle of it."  And I'm not entirely sure it's mentioned in any official 5e sources, but it might.  I first saw it in the DMs advice chapters of the d6 Star Wars game.  The first scenes in A New Hope is literally a text book illustration of in media res.  The Tantive IV is under assault and R2 and 3P0 are reacting to it with very little control over the circumstances.  The prisoner starting point of the Out of the Abyss is a bit more static when it's presumed the PCs have been captured, but still stuff has already gone down.  So nothing wrong with starting your game in the middle of the action.  I actually can't think of a style of story (crime, war, diplomacy, exploration etc) where in media res starts are inappropriate.  Railroad is just synonymous with "bad word" if it can be applied to an entire campaign.  Like it or not campaigns are established on something, call it a railroad, or call it a foundation or germ/seed.

As far as your specific thinking, you got two set ups: the botched job or the boss playing them for patsies leaving them to die in a set up.  I wouldn't do the botched job, because in reconstruction of the botched job, blame could easily fall on a single character and criminal players would likely see "making it right" to their boss would involve proactively sanctioning the player at fault.  You don't want to encourage inter party conflict via events no player had agency over.  The set up works, and is a common start to criminal campaigns.  The players are sent on a wild goose chase/red herring because they're low level nobodies, so the boss tips off the authorities while the boss's actual "A Team" goes after the actual score.  As a DM your challenge is how do you give your players an out from what is supposed to be a kill box.  You'll need to have some sort of environmental factor or maybe social/NPC factor that gives the players an out before they take the fall, a factor that the mastermind was unaware of.  From there, the players have to negotiate with 1.) likely being fugitives from the authorities they just fled (and maybe killed). 2.) Figuringout they were set up 3.) orchestrating payback.  This actually sounds like an archetypical thieves guild type starter campaign.  Mechanically, the set up functions as tutorial to combat as well as creative thinking to extricate themselves from a fight they aren't supposed to win, follow up investigative steps tutors role playing and skill checks.    

I don't know of anything at all in the game that players HATE more than failing.  Players want to be the heroes of the story, the main characters, they want to feel like their contributions are important, and their choices meaningful.

If you want the characters to be criminals, working for criminals, and getting screwed over by a boss who wants to kill them, might I suggest Cyberpunk, or better yet, Shadowrun?

Quote from Geann >>

I don't know of anything at all in the game that players HATE more than failing.  Players want to be the heroes of the story, the main characters, they want to feel like their contributions are important, and their choices meaningful.

If you want the characters to be criminals, working for criminals, and getting screwed over by a boss who wants to kill them, might I suggest Cyberpunk, or better yet, Shadowrun?

Absolutely, criminality and the theme of patron betrayal that often comes forth in criminal stories have no place in D&D ... despite the existence of thieves guilds, a language/argot specifically developed to talk about criminality, character subclasses described as the sorts often utilized as assassins, a character subclass actually called the assassin.  I mean the classic AD&D setting Lankhmar City of Adventure is utterly untenable in 5e and listing Fritz Lieber in the PHB handbook page on "inspirational reading" was a joke on those looking to play D&D as something other than a performance of virtuous nobility.  :-/ 

Edgerunners are Adventurers and Adventurers are Edgerunners.  Earthdawn and Shadowrun were in the same universe for a reason.

If you really weren't joking, read some of the constructive comments to the OP.  There are ways of enjoying the game besides the mode you're outlining.