You can pick up a small policy to cover just the contents in the storage unit. Can be as limited or as broad as you like.
Ideally, though, since you don't own a home, buy a Renters Policy to cover your belongings where you live and it would also cover liability.
That coverage is usually very inexpensive and can come in really handy if: There's a fire in another apt unit and your stuff gets burned; the sprinklers go off and your stuff gets soaked; a guest leaves and falls injuring themselves; stuff like that.
Then, as suggested above, you can pick up a small rider to cover the belongings in the storage unit.
If you go for the storage unit belongings alone, you'll likely pay more for that piece than you would purchasing it as an add on
One quick note: expect pricing to be a little bit higher because of the potential for wind damage in Houston. Also, potential for Flood depending upon where in Houston, is always a concern but more so now since the recent flooding. The Flood aspect of coverage is almost solely dependent upon where, in the building, your belongings are placed. 2nd story and higher, you might consider not pursuing or getting the coverage but with a high deductible
