No, they assume no responsibility whatsoever. They do offer insurance, but we have renter's insurance already, as well as WP Dolle insurance for my Steadicam gear. Filing claims now, so hopefully everything will be reimbursed.
Be warned if you store things at one of these facilities. The thieves followed someone in through a locked, coded gate, pried the deadbolt open, not the lock, and ransacked the unit. It was during the daytime, normal business hours, and there's video. It's very low res and from far away though, so it seems like it probably won't help much.