Archiving would happen in the background. If you are running background tasks based off of traffic, it could well be these are not moving due to lack of traffic. You can try to change this to CRON by going to ACP -> System -> Advanced Configuration.
Ideally though, our recommendation here would be to move to a hosting provider that does not limit your databases to be this small. In due time, you will quickly reach a point where you need to archive more content and more content, etc...