Your oral contract is enforceable, but obviously it's more difficult to prove than if you had a written one. It would help if you had emails or texts to document any of the contract.
Abandoning the job is a violation of CA Business & Professions Code, and your contractor could face criminal penalties, discipline and licensing problems depending on their license and if they misrepresented their licensing or insurance status. If they do have a license (check the link below), you can file a complaint with the CSLB (also linked below), and also, if they're bonded, you can file a claim with their surety (you need to do this quick, the deadline is very short). These actions may put some pressure on them to respond to you and either finish your job or refund some of your money, and if that doesn't work, you can sue.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
If this was a home improvement contract, the contractor has an obligation to have a written contract with you. In addition to contacting the Contractors State Licensing Board, you may also consider making a claim against the contractor's license bond. Information concerning the contractor's license bond will appear on the Contractors State License Board's website. Just look up the contractor and the information will appear. You can write a letter to the bonding company and request payment.
The San Diego County Bar Association can assist you with finding a lawyer who can advise you concerning your situation. A link appears below.