Is there a way for someone who is not a monk to catch a projectile, similar to the monk's Deflect Missiles feature?
Let's say there are some bad people with bows. Can I ready my character to catch an arrow coming their way?
Just find a pair of Gloves of Missile Snaring and you are set:
These gloves seem to almost meld into your hands when you don them. When a ranged weapon attack hits you while you're wearing them, you can use your reaction to reduce the damage by 1d10 + your Dexterity modifier, provided that you have a free hand. If you reduce the damage to 0, you can catch the missile if it is small enough for you to hold in that hand.
Tell us a little bit more about your characters intent. Why do you need to catch an arrow in your hand - to string in your own bow and return fire?
If I were DM'ing, I'd ask what you wanted to accomplish - and if your character had a shield proficiency, I'd recommend discussing a series of actions that could result in you using a wooden shield as a reaction to stop the projectiles - and on your following turn, you could pull an arrow out for whatever purpose you had in mind.
While it certainly wouldn't be easy, it would be the best way I can imagine of doing what you want short of supernatural intervention.
It's not necessarily a 5e solution, but in 4e Oriental Adventures (I believe) there was an option to design your own martial arts. E.g. did you want a hard martial art like karate or a soft martial art like tai chi -- these styles defining the particular focus of the techniques -- are you offensively oriented or defensively oriented. Additionally you could pick various things that your variant could do (or use the preselected options).
It was based on 'buying' feats at various levels and building up your list of skills and maneuvers. Basically at rank 1 you learned some basic kicks/punches/blocks and as you got more experienced you could add various more advanced techniques. The techniques were all mundane and didn't require anything like Ki to operate.
While the rules were in that set of supplements, it wasn't so overpowering that it couldn't be merged in with a 'normal game', where not every character is surrounded by martial arts constantly.
Additionally, it was open to anybody (no class restrictions) -- though, Monks, would get the cool, super powered options. Everybody else could learn some mundane version of skills...even though, some could get somewhat good.
This way everybody could have an option whereby they could learn a skill like catching arrows (and it being a codified method based upon real rules [the caveat being that it's 4e], not being totally a house rule.)