The only person that can tell you definitively whether or not the storage facility is responsible for your damages is the judge.
Now as to whether there is any chance of your claim succeeding, your claim will only succeed if you are able to show that the facility had a duty to protect your belongings from rats. The first place to look to determine your claim's viability is the agreement that you signed with the storage facility. If a review of the agreement does not make it clear to you, you should show it to an attorney to obtain an informed opinion.
Good luck in court.
~ Kem Eyo
The above answer is a general explanation of legal rights and procedures. It does not constitute legal advice. Nor does it establish an attorney-client relationship between the individual posting the question and the attorney providing the answer.