24/7 Customer Support. StorQuest Self Storage in Claremont, CA offers both outdoor drive-up access units and indoor units. Sizes range from 25 sq. ft. to 300 sq. ft. With over 200 storage units to choose from, we guarantee you’ll find the perfect space for your belongings. Our facility is equipped with 24-hour video surveillance and computer-coded gate access. Our location is also equipped with storage units for businesses.