Legal Question in Construction Law in California

Sorry to say, doing contractor work without being licensed is illegal, and the courts will not enforce your agreement - not for the reason that it was verbal, but for the reason that you are unlicensed.

You do mention agreeing to work for an hourly wage. This may suggest you were working as an employee of the owner, rather than as a contractor. There is an exemption in the contractors state license law for employees (at Business and Professions Code section 7053). However, you would not be considered an employee unless you worked solely for wages, you were not operating like a person in business for himself, and you had no control or discretion over how the work is performed in terms of the final results. Agreement on a lump sum ($7,000) is contrary to being an employee working hourly.

I think that although the owner might have a moral obligation to pay you, I don't think there is a legal obligation that you could enforce.