Based on current bitcoin technology....
1) Public wallet system, stored in a P2P network
2) Block chains performing transaction validity
a) correct balance to transfer
b) valid destination and origination wallets
3) User creates account, private client issued to user.
a) Private wallet shows account balance
b) Provides authentication mechanism to initiate transactions
4) Miners block chains help ensure the security and functionality of the bank
a) Miners are rewarded with a nominal transaction fee for banking operations
Obviously a lot more could go into this, but this could help further the bitcoin cause if it is feasible.
Sounds like you are describing the existing Bitcoin network.
yeah, you might aswell just send your bitcoins out into the void and then load up a second client when you want to grab them.