MySQL has a way to store tables in CSV. It has some pretty serious limitations, but it sounds like your requirements demand something with some pretty serious limitations anyway.
I've never set up a Rails project that way, and I don't know what it would take, but it seems like it might be possible.
HSQLDB seems to work by storing data on disk as a SQL script that creates your database. It records changes in memory and a log file, and when you shut down it recreates a single SQL script again. I've not used this one myself.
HSQLDB doesn't appear to be one of the supported databases in Rails. I don't know what it would take to add support for a new database.