It could be a problem with your template or a conflict with a plug-in.
Have you tried switching your site to one of the WP default templates and then rebuilding the permalinks again? Does that help?
You could also try removing the current htaccess file and let WP generate a new one by resetting your permalinks. (If you have a lot of customizations, save a copy).
If you think it is an issue with the databases, can you roll it back to a point before the 404 errors started happening? (You find that in your Dreamhost Panel, under goodies > mysql databases). The database and the WordPress php files - and your media and images - are hosted in two different places.
Can you access your posts and media from the dashboard? If so, export your site content before you do anything with your database, so you can import it into a clean WordPress if you need to.
Also, make sure your site hasn’t been hacked. You can ask support to run a scan for you if you think that might be the case. Open up a support ticket.