Tenant was evicted, we placed all belongings in a storage unit since we knew

tenant was not retrieving their items by the thirty days we were required to hold

their items. Turned over the storage unit to tenant within the thirty days, had tenant

sign a letter that once she took over the storage unit our relationship was done.

Now four months later the tenant is calling claiming items are missing. The tenant

had access to the storage unit for the last four months.

What are my legal rights versus the tenant's rights?

You should always have them sign an indemnity and hold harmless form. This is the only way to keep weasels like this at bay.
If the value of the items they are claiming are significantly high like their grandmothers turn of the century china collection valued at $20,000 I would tell them to get lost. If they are missing their blender tell them tough luck and have them come after you with their lawyer costing them a Ninja Blender hundredfold. They won't.

You don't have any special rights in this situation. But neither does the tenant. It's just plain old suspected theft. I would suggest to the tenants that they file a police report.  Don't assume anything.  Storage areas have plenty of crime. If the tenants make a report, the police will investigate and most likely contact both the storage unit manager and you (since you opened the account).  Just like any other suspected theft, that's how it's handled.  The tenants need a police report to bring criminal charges and/or to make an insurance claim.  

My guess is that they won't contact the police to file a report.  If they do, my second guess is that the investigation will be inconclusive.  Let the process play out on this one.