No, any more than its illegal for a storage company to go into the lock-up you rent off them if they need to make sure there's nothing flammable or dangerous stored there.
Then they would have to pay you for damaging your lock. Every storage facility insists on you using your own lock (they may sell you one, but it must be yours) because they do not want to be stuck with the liability for theft. You normally agree contractually to provide access if they want to inspect (and, of course, to not do what they forbid you) but they can't just walk in as they please as you suggest.
Ditto for MS, and there's even a question if demanding that of users is legal. For that matter, even Google's scanning of inboxes is technically illegal in Europe because they will not have had advance permission from non-Google senders for that (and that's not my opinion, that was the statement from Data Protection people in 3 different countries when asked off the record - this is not actively pursued because of politics more than law).