UBB.classic stored its data in flat files - 1 per member, 1 per topic in each forum, etc.

UBB.threads v6 & v7 both mainly store their scripts and config on disk. The topics, members, etc are stored in the DB.

I know that v7 does have a few files generated during operation (sessions, templates/compile). Keep in mind that enabling uploads and logging will increase the number of files that are required to operate your forums.