We've been playing my homebrew campaign for 6 months now and my players are getting a bit fatigued by their character's abilites.
In short: The spellcasters are anoid they can 't deal decent damage with weapons and the fighters are jealous of the spells the spellcasters have.

In a way, I wanna tell them that that's how their characters are build and that you shouldn't have a character that ticks all the boxes. On the other hand I do want them to still have fun and enjoy their characters and the campaign.

So I'm looking for some things I can give the spellcaster which would make them a bit more tougher, but not too tough so the barbarian feels useless. And I'm also looking for ways of the non-magic users to have access to some magic in a way.

Any items or ideas I could use to create this effect?

Multiclassing exists.

Why not at a natural pause point throw in a one shot set in your games past and let your players roll brand new characters for it that they can have a dip and see if the grass is truly greener before you change everything up for something they might not care about.

If you are 6 months into a campaign, this is might be a bit late but there are subclasses that are designed to give non-magic classes limited magic abilities, and spellcasters the ability to not be completely squishy. 

For example:

• Hexblade Warlocks get proficiency with Medium Armour, Shields, and Martial Weapons in addition to the Warlock spell list
• Arcane Trickster Rogues get the ability to cast a small number of Wizard spells from Level 3 onwards
• Eldrich Knight Fighters are basically the same thing as Arcane Tricksters, but based on a Fighter rather than a Rogue
• There are also quite a few Monk subclasses that give access to spell-like abilities

Multiclassing is also a thing, and for a character with high charisma a single level of Hexblade will give casters the ability to stand up in melee a bit more, or non-casters access to some basic spells.  Just remember that the stat requirements for multiclassing are there for a reason - there is no point in letting your INT 8 Barbarian multiclass to Wizard only to discover that their spells are completely useless in combat. 

Alternatively - if the issue is that your players are just a bit burned out after 6 months of using the same abilities all the time, why not run a one-shot and let your players roll up some new characters to try out?

Magic initiate feat for the fighters.
Let casters swap out their cantrips to take green flame/booming blade.

Let them respec their subclasses — possibly after tasha’s comes out since it may have even more options, like that barbarian spellcaster and I think a red one blade singer.

Also, I agree with your instinct to explain to them (nicely) that you can’t have one char who does everything. It would unbalance the game completely. And that D&D really hasn’t quite cracked the code on a good, balanced gish; they’ve got to go one way or the other. 

What level are the characters in question?
Which optional rules are you using in your homebrew?
Did you want suggestions for standard magic items or will you homebrew these as well?

I agree, let them multiclass. It's super easy for the fighter to start casting spells -- just take a couple of levels of Cleric, Druid, Sorcerer, Bard, Wizard, Warlock.

Vice versa for the casters.

Let them switch subclasses if one of the subclasses allows for spellcasting, like arcane trickster or eldritch knight, you can come up with some sort of narrative way for it to make sense. Or just multiclass like others have said.

In all seriousness, maybe play another system?

A classless game system might be a better fit. For example, Savage Worlds, GURPS, Fate Core.

My question is: is everyone new? If the campaign has been running for 6 months, is this the first 6 months they have played (or close to it). 

People get bored. No shame in it - I am surprised people manage to keep a character for 2+ years and are happy about it (despite that I tend to play a wizard over and over ...). Your players may have picked what they initially thought was good and now want to try something else. No shame in that. 

As sardonicmonkey said - either try a one shot or play a "meanwhile" where you go back a few months and play a different aspect of the story from another point of view. Additionally if they are getting bored in 6mths maybe aim for shorter campaigns - so that they can get through it and change characters etc without you feeling like you wasted your time and never get to the end of a story. 

Of course it is possible that the players will always get bored and want to be jack of all - then multiclassing works or shift towards different subclasses (or just play a bard ...)

I would let them change their subclass if they want to be a different one, and also allow multiclassing and recommend feats that give spells. If you're okay significantly increasing their combat capabilities you could also give them some magic item that would let the martial fighters cast and casters use martial weapons. You could give them wands that can let them cast spells, and the casters could get magic weapons, possibly ones that give them proficiency in armor/weapons (bracers of archery, elven chain, etc).

If the martial characters want to cast spells, they could choose their subclass/multiclass into another martial class that can cast spells. They could multiclass into Battlesmith Artificers, which do a good job of blending spells, support, and weapon attacks. For Barbarians the only real option is the Wild Magic Barbarians, and they more have magical abilities instead of spellcasting. Fighters have Arcane Archers and Eldritch Knights as subclasses, but they're kind of disappointing subclasses, but your players may be fine with that. Monk subclasses that can cast spells are Way of Four Elements, Shadow, and Sun Soul, but all of them are pretty lacking in casting abilities. Paladins are very satisfying casters and martial combatants, so you could allow a bladepact warlock or fighter become a paladin through swearing an oath or some other story element you design as the DM if you want to let them change all their class levels, otherwise they could multiclass. Rangers are also casters and martial combatants, and you could let a druid, fighter, or rogue. There are Arcane Trickster Rogues, which are just as satisfying spellcasters and martial warriors as paladins, especially if you build them right (booming blade/green-flame blade and shadow blade). 

Bards have the College of Valor and College of Swords. Clerics have War Domain clerics and a few other domains can be good at melee fighting in specific circumstances (Arcana, Nature, and Tempest Domains). Druids have Shillelagh and Circle of Spores that can help in melee combat. Sorcerers don't really have any options to fight. Warlocks have Hexblades and Pact of the Blade to fight with melee/ranged weapons. Wizard have Bladesingers, but they're not very balanced, but that might not be a problem for you.