[damien_ls]

It is technically possible to use a database from some other hosting provider. However, this will almost certainly cause your site/scripts to run slowly because you'll introduce additional latency between your webserver and your database server.

The other issue that you may encounter is that most decent hosts will restrict database access to the local server rather than making it open access, since this is a good idea from a security point of view (i.e. less chance of people gaining access to your databases to mess them up).

As a side-note, we don't limit MySQL database size - it's just part of your overall space quota (e.g. you get 1GB for your website, emails, logs, database etc.). This is quite common, so I don't think you'd struggle to find hosts with a similar approach.