First off, just because no cash changes hands, doesn't mean that a service is "free." There is only a contract if there is an exchange of value. For example, if I give you my bicycle, there is no contract. If I give you my bicycle in exchange for $1, then there is a contract.
The data storage facility in this case is not paid cash money by you, but rather by your agreement to surrender an item of value, likely in the form of allowing metadata about your interaction with their system to be bundled and sold to third parties.
So there IS a contract. The provider must exercise due diligence in the course of handling your data as laid out in their terms of service. If the terms of service was properly presented at the inception of use, then the user has little recourse if the failure by the provider is anticipated and described in the TOS.
The answer is basically that it isn't the "free" part that governs here.