Is it possible to enforce a verbal agreement between family members

Yes it is possible to enforce agreements between family members. While certain agreements, are required to be in writing, such as the transmutation of community property to separate property, -- here, where the agreement is between a father and son for the payment of a monthly income by the son to the father in exchange for deeding the father's multifamily dwelling to the son, an oral agreement may be enforceable as a result of the performance by the father of his part of the contract. I believe the son would be "estopped" from claiming that the agreement was invalid because it was not in writing, according to general principles of equity.
The main difficulty for the father in enforcing his rights will be a question of evidence, i.e. whether the father is believed or not. But as a matter of law, it appears that the father could enforce the oral agreement if the court believed the father's evidence.