You don't mention where you live or your age, so YMMV. If you are in the USA, contact a local insurance agent (or a few to shop for price) and inquire about a Named Operator policy. Such a policy is for people that don't own cars.

You can purchase minimum limits for your state - the policy will cover you in CA automatically. Named Operator policies are generally cheap, and you can probably even cancel the policy after you get back from your trip - getting a refund of the unused portion of the premium.

Check on cancellation fees for the company from which you buy, but going this route will probably be significantly cheaper than trying to buy anything from Avis.