When buying a used car in California, does the buyer have any recourse if the car renders inoperative within 90 days of purchase and there is no written contract.

You can always try to sue, if you can prove the previous owner was aware of the defect. Also was the bill of sale marked as is, if there was no notice that the sale was 'as is" the argument that an implied warranty may apply.

Generally, used car sales from private parties are considered "as-is" sales, with no implied warranties. And the seller's statements that "the car runs well", "in mint condition", etc. are not considered to be warranties. Something like "70% tire tread left" or "two-year-old battery" are considered to be warranties of that part of the car.

Did you have the car checked by a mechanic before you bought it? (Not that this will help in your case now, since he either said the car was fine, in which case you are responsible for repairing things that break later, or he said that it wasn't, in which case you knew about the issues when you bought the car, and thus are responsible for fixing them.) If you didn't, you have probably just had a very expensive lesson in life.