Generally, renters insurance will cover your stuff in storage units, but at a lower limit. Items in a self-storage container are covered by renters insurance up to 10% of your policy limits. If your policy's limitations are insufficient to cover at least 80% of the value of the goods in the unit, you might want to consider purchasing extra storage insurance from the storage provider.
The information in this answer is general in nature and is not legal or insurance advice for your specific situation. Policy language varies carrier by carrier; your declarations page is the final word. If you're unsure, call your agent before you need to file.