I'm wondering if you could make a really simple social network without using a database. (Well, you'd probably need one for login information, but other then that...)
Each users messages are saved in a text file. The people they follow are also saved in a text file. When they want to see messages they request the messages from each user one at a time, and reassemble them on the other end.
The client requests the messages a small number at a time until it reaches messages it has already seen.