I think the two most storage related things to monitor for the server would be the location storing the database and the backup storage itself. It would also be handy to have an email notification if the storage containing the database is running low.
Maybe a GUI object, something like this would be useful too?
Then, if for some reason the email notification didn’t go through, when checking the server manually, it would be simple to keep an eye on the storage. I suppose this should only show up in the server admin accounts…