If you have a written agreement concerning storage, look for a clause that says the landlord is not liable. Such clauses are enforceable. If not written agreement and it is space controlled by you, the landlord will not be liable. If it is space controlled by the landlord, that will be governed what is called a "bailment." In such case the landlord is required to use ordinary care to safeguard your property in his possession or control, but does not guarantee it safety under all circumstances.
This comment does not create an attorney-client relationship. The law and its application by the courts is constantly evolving and changing. This discussion is not to be taken as a definitive guide, and should not be relied upon to determine all fact situations. Each set of facts must be examined separately with the current case and statutory law analyzed and applied accordingly.