Yes, normally if you were not at fault in an accident you can place a claim through the at-fault person's insurance even if you were without insurance yourself at the time of the incident. The status of your insurance is not usually taken into account by the other party's insurer when they are determining fault and if their policyholder is responsible for your damages.
You should have been cited by the CHP, or other responding law enforcement agency, for driving without insurance. If you are caught driving without liability insurance in California you can be charged hefty penalties. Violators can face up to hundreds of dollars in fines for a first time offense. Also, if you are involved in an accident and do not have liability insurance, your license may be suspended whether you are at fault or not.
For a quote on California auto insurance, so that you can place it on your vehicle and be legal, click here.