Let's find out what the backup tool is trying to "do".
Open the backup page
click the backup button
before you click Backup Now in the right-hand sidebox, note the bold text there ... it's a list of command-line parameters.
Copy and paste it here.
But be sure to replace ONLY your database-password with *****.

The database name listed in the parameters is the one it's attempting to do a backup on. Is that the same database that your host says there's no "files" table in? It shouldn't be trying to backup a "files" table if there isn't one ...