A BitCoin Bank would be a very simple concept. It would be a Safety Deposit Box system for the .dat file accounts. The bank won't know who the customers are or how much are in the deposit boxes. They just charge a fee for keeping the .dat files safe. A good bank would have mirrored backed up boxes across the world and even possible physical .dat files in the form of printed binary for emergency recovery. (ie. EMP pulse takes out all the electronics world wide or something of that nature). You could recover your physical printed binary file and start typing the data in anew to recover. It would be a pain but hey if there is a million buck in there I will type for you.

The second service provided would be a type of holding house. You pay for something, they take your money, tell seller to send product, when you receive it, they clear the transaction to the seller.  Trusted Man in the middle. A lawyer-less Escrow account in away. Of course a level of anonymity would be stripped for this to work. PGP keys could work for this. But the Keys Servers could be attacked to disrupt transactions. Unless a system of PGP P2P Keys for sending messages is accepted. The "Bank" would have to have the Public Keys for each transaction and be able to generate messages to each party involved. I guess the the "Bank" could set up its own on account opening and provide a unique private(sort of sense it would need to be given out) key to each account holder for communications.