SQLite definitely does not require a filesystem interface, or even standard OS interface. It does require some kind of storage, though, but that could be direct flash, RAM, etc.

I have spent a lot of time with it on ARM Cortex-M-class devices on small RTOSes or even bare metal, and you can adapt it to almost anything by plugging in new VFS interfaces (http://www.sqlite.org/vfs.html).

It does require at least 10s of kb of RAM as a heap to do pretty much anything, as others have stated.