Can you describe the units better? Are these apartments that happen to have a storage closet off the balcony? Or is this a storage-only facility?
If it's apartments, I'd think the tenant's renter's insurance should have to cover it. A normal lease would say landlord isn't liable for damage to personal property, even if the cause is not the tenant's fault (like ruptured pipes in the upstairs apartments, etc.) that the tenants are advised (or even required) to get renters insurance for any damage to their personal property.
If it's a business where people pay to store things in storage spaces, and the units had mold, in that case it would seem the business insurance should cover it. Again, there should be something about damage in the contract.
So, I think you should start by looking at the contract. Next talk to insurance company about coverage.
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