Is it possible to create a system for 'shared wallets' on top of bitcoin? Is it possible to build a system on top of bitcoin, where:


Random people will deposit money in a wallet X.
Nobody has access to that wallet.
After a few months, those users will vote in other wallets for the money in wallet X to go.
The money in wallet X is distributed between those wallets.


Is it possible on top of bitcoin, or would it need the design of a new system?
Yes, it's called a Distributed Contract or a Smart Contract. 

Projects like Ethereum are aiming to make this much more of a reality. Currently they're a bit of a pain to write.