Magic Jar is most often encountered as a trap of an enemy spellcaster, but the fact that it's a spell that players can learn makes it useful to think about what that would look like for an adventuring party traveling with its caster in such a jar. Or, I suppose, traveling with a jar that their friend was sucked into! Poll up top, leave a vote and post your reasoning/concerns!

Without worrying about all the rules about how you attempt to leave the jar, or what happens once you do, let's focus on what you can do while riding around inside it.

...While your soul inhabits the container, you are aware of your surroundings as if you were in the container's space. You can't move or use reactions. The only action you can take is to project your soul up to 100 feet out of the container, either returning to your living body (and ending the spell) or attempting to possess a humanoids body.

...

Meanwhile, the possessed creature's soul can perceive from the container using its own senses, but it can't move or take actions at all....

 

Looks like whether you're the original caster, or a poor rube that got sucked into one, you have the same ability to perceive your surroundings as you had in your body. No downgrade to standard vision or limited radius to be found, if you had Devil's Sight or Darkvision or whatever, you still have that. 

A victim can't move or take actions at all, and a caster is only slightly free-er, not being able to move or take actions other than one specific action to try to possess a humanoid. But what does it mean to "take actions"? This'll take a little bit of a more thorough breakdown and a little DM decisionmaking...

Now, the pedantic among us might be inclined to read "take actions at all" to encompass pretty much everything short of thinking or perceiving surroundings, but 5E has used this phrase (or a very similar one) before to provide us some clearer RAI guidance in PHB Chapter 9: 

On your turn, you can move a distance up to your speed and take one action. ...anything that deprives you of your ability to take actions also prevents you from taking a bonus action...

Other minor activity, like blinking, picking your nose, tapping your toe, etc. isn't really considered "actions," but rather "other activity", unless you're doing a lot of it, or there's some other special obstacle that makes it not trivial. 

Your turn can include a variety of flourishes that require neither your action nor your move.

You can communicate however you are able, through brief utterances and gestures, as you take your turn.

You can also interact with one object or feature of the environment for free, during either your move or your action. For example, you could open a door during your move as you stride toward a foe, or you could draw your weapon as part of the same action you use to attack.

If you want to interact with a second object, you need to use your action. Some magic items and other special objects always require an action to use, as stated in their descriptions.

The DM might require you to use an action for any of these activities when it needs special care or when it presents an unusual obstacle. For instance, the DM could reasonably expect you to use an action to open a stuck door or turn a crank to lower a drawbridge.

But even if "other activity" stuff isn't really "taking actions," are there contexts where not being able to "take actions" suggests you would also be prevented from any "other activity"? Kind of! Let's look at Conditions, from PHB Appendix A. The standard we'll start with is Incapacitated, a condition wherin you "can't take actions or reactions." Can you move while unable to take actions or reactions? Yup, because see other conditions that incorporate Incapacitated + can't move, like Paralyzed. Can you talk while unable to take actions or reactions? Yup, because Paralyzed again treats "can't speak," "can't move," and Incapacitated/"Can't take actions or reactions" as three separate effects of the condition.

I think that Paralyzed, and other conditions like Petrified, Stunned, and Unconscious, are enough for me to feel confident that "can't move or take actions at all" is "can't move" + Incapacitated, and does not include any jar-anatomy-or-mental "other activity" that a character might be able to muster up.

It is unfortunately worded in a way that RAW (perhaps not RAI?) prevents the jar from making any Search actions in combat to spot hidden creatures, though perhaps not so strictly that they aren't still permitted the ability to make Perception checks or have a passive perception score in and out of combat? See also History, Arcana, etc... the jar is still able to see, hear, and think, so I don't know why it shouldn't be permitted to use those skills, short of actually taking actions in combat. 

And if the creature had telepathy before going in (either racial, or as a GOO Warlock, etc.), I see no reason that the jar should not be able to speak to others and hold conversations, or even make Persuasion checks. Being able to "communicate however you are able" is a prime example of "other activity" that is short of actions in PHB 9, and while a jar may or may not have the anatomy to speak (it doesn't need anatomy to see/hear/smell/etc., so that's not necessarily fatal to the idea that you should be able to talk...), a creature with telepathy seems like it should still have free use of that. 

So what do you think... character in a jar, would you let it talk telepathically and participate in skill checks, short of actually moving and taking actions?

What about class pets and other features that allow you limited ability to direct or control the pet without an action or bonus action? If you can communicate with the pet the way you normally would to exercise that, can you continue to do so while in a jar? 

Could be a lot of fun. 

As a DM, I'm tempted to let them get away with any kind of thing that doesn't require them to interact with the environment. So if they can already communicate without flapping their gums, then go for it. 

That does limit them to things like passive perception checks (can't rummage through the desk looking for stuff), and I'd be tempted to allow them to even perform activities that are purely mental, such as recalling facts about a monster that is currently attacking the party. 

Imagine if you also had a familiar?  It's a pity you have to use an action to use the familiar's senses, but you could still mentally command it (no action) to do stuff (like pick up your magic jar and take it somewhere so you could see better). 

To me, the drawbacks and risks of Magic Jar make it dangerous enough to use that adventuring while inside one is not exactly game-breaking. It does make for a nasty infiltration tactic though, if you do allow familiars to be commanded from the jar.  
Go into Jar, Command Familiar to take jar on a scouting mission, leaving your body safely with your friends. Try to possess someone important at your target location.  If successful, go around ordering guards to change patrol routes, opening doors etc. Keep your familiar within 100 feet.  Then, when your friends catch up, go on a rampage until the enemy kills your current body. Return to Magic Jar. Rinse, repeat. (Just make sure the Familiar and the jar are never near any combats, which is a lot easier if you are Pact of the Chain or equivalent).

It's not risk free. Familiars are squishy, and your magic jar is basically defenseless. And if that jar is destroyed and you're more than 100 feet away from your body, you die (although you probably have a friend who can raise you, since you're 11th+ level).

Absolutely! 

I think that there's potentially some (arguable) ways to build yourself a tank body using Magic Jar, by either True Polymorph the jar into a friendly Clay Golem or the like (not that you control directly as your body, but rather which your senses and mind ride within, while telepathically communicating with the loyal golem, and waiting for a moment to possess out of it), or just having a summoned/created creature swallow or incorporate the jar in a way that protects it (Polymorph on your familiar should let it merge its possessions into its new form, preventing the Jar from being destroyed until the form is killed).

But those are all edge cases that the DM has other off ramps to disallowing (like ruling through DM fiat that the jar stop functioning when it loses the form of a jar?), at its core, I agree that a player in a jar should be able to meaningfully speak (at least telephathically) and participate in skill checks that don't require a body while they're a jar, and being carried around by their Familiar or Steel Defender or whatever. 

Interesting. 

I agree there might be some barriers to those ideas that might be tricky to work around. 
Since you can't cast spells while in the jar, you'd either need to cast the True Polymorph first , after which, the jar is no longer eligible to be the material component or you'd need a friendly spellcaster with True Polymorph to cast it for you (which raises the question -- to whom is the golem loyal? and how are you commanding it?) 
I guess you could cast it while possessing a different body. That would work. Whether a DM allows it to still be a jar after that though.. That's more difficult to answer. I could go either way. Depends on the style of the game I'm running, and how the other players react. 

And if you concentrate on TP long enough to make it permanent, then you no longer control the golem. Since it wasn't created using the regular means, it might not regard you in the same way as a normal golem does.  That's also unclear. 

BTW -- true polymorph for Object to Creature says: 
"You can turn an object into any kind of creature, as long as the creature’s size is no larger than the object’s size and the creature’s challenge rating is 9 or lower".
That is one LARGE crystal magic jar you need to lug around when it's not in the form of a Clay Golem.

I'm a little concerned about Polymorph's melding of gear. The way I read that implies that the gear vanishes and has no more game effects while the poiymorph is active. I might be tempted to say that the jar is no longer something you can access, and certainly if you were in it, you could not see anything, since it's not really there.  Same with a familiar eating the gem. You can now see the inside of the guts of your familiar :)   Again, I'd be more lenient in some styles of games than others. Myths and legends are rife with stories like this so it could be cool. Even more so if it's an evil NPC the party can't quite seem to kill. 

Take it to the next level : make the magic jar a large crystal container that you put your original body inside, completely protected. Then cast TP to turn it into a Clay Golem, and your original body vanishes, melding into the golem ;)  Hilarious.