Well - the size of the backup file obviously depends on what you define as fairly small and how large your database is. I wouldn't call my database backups fairly small, tbh
phpMyAdmin is not a good tool to do database backups - especially not to do restores. If your hoster does not provide that functionality (and most do not), I recommend MySQLDumper:
http://www.mysqldumper.net/
Or, of course, if you have shell access, you just can do it via a command line cron.