The core feature of a social network is the data that relates one user to another. If you want to operate without server side code to handle it then you will need to move that logic to the client and share it with other clients.
DOM storage will let you store large (but not huge, so you get limits on the amount of data that can be stored) amounts of data on the client.
(Although it will be freely editable by the user of the browser so security it out the window)
Then you need to share that data between clients. That means peer-to-peer. Even then, I think you'll need some way to perform browser discovery so the browsers can communicate with each other and I suspect that will force you onto having a central directory server.
If you can deal with all of that, then you could probably implement a Napster-like system for searching the data in the social network.
So possible? Almost. Practical? Probably not.