300mb database? how busy/big is your website? sometimes spam comments can take up a ton of space. you can clean out the comments via phpmyadmin if you can't get into wp-admin. 'antispam bee' plugin helps a lot to prevent spam.
Can you contact your host and access the database via cpanel/plesk -> phpmyadmin? while in there, you can "optimize" the tables which will remove overhead (which can take up a lot of space: it's basically cache).
Another option if your site is overly busy is you can disable andor remove post revisions. they save every alteration to every post, which on busy sites can take a ton of space.
If your site is big and awesome, and the 300mb is actually real data, just upgrade your hosting (though I've never seen a host restrict mysql size.. Usually you see 300mb with sever storage or bandwidth, are you sure it's not one of those?)