Sounds like an insurance job - you need to check the rental agreement and see whether you can claim against the owner's insurance. This kind of thing is why indoor dry storage is more expensive than renting a garage.
Caveat, not a legal person at all, just my experience.
When I rented my house out a few years ago, my tenants did similar and stored their furniture in our garage which was a separate outbuilding. The garage let water under the door and some of their furniture was damaged. They wanted to claim through us, but were firmly rebuffed through the agency and our insurance company that they had rented a garage which does not have to be watertight to be fit for purpose.
We (living overseas at the time) did not know they wanted to use it for storage.
My guess would be that it might hinge on whether you have their assurance that it would be watertight in writing, as part of your contractual agreements on the rental.