You need to continue to pay for the storage unit or clean it out and close it. Otherwise, there is a chance all of the property in it could be taken

If the deceased person was the only one living in the rental, you will want to negotiate the right amount of time to get all of the belongings out while grieving, but avoid paying extra rent. Depending on the lease, a written notice of death may give 30-day notice on the lease or not affect the lease. However, even if the lease is not ended by the death, most landlords will be willing to work with you to end the lease sooner so that they can get the unit rented sooner and you don't have to pay for an empty unit. Landlords may also help you protect the property.