Non-orthogonal multiple access (NOMA) based fog radio access networks (F-RANs) offer high spectrum efficiency, ultra-low delay, and huge network throughput, and this is made possible by edge computing and communication functions of the fog access points (F-APs). Meanwhile, caching-enabled F-APs are responsible for edge caching and delivery of a large volume of multimedia files during the caching phase, which facilitates further reduction in the transmission energy and burden. The need of the prevailing situation in industry is that in NOMA-based F-RANs, energy-efficient resource allocation, which consists of cache placement (CP) and radio resource allocation (RRA), is crucial for network performance enhancement. To this end, in this paper, we first characterize an NOMA-based F-RAN in which F-APs of caching capabilities underlaid with the radio remote heads serve user equipments via the NOMA protocol. Then, we formulate a resource allocation problem for maximizing the defined performance indicator, namely network profit, which takes caching cost, revenue, and energy efficiency into consideration. The NP-hard problem is decomposed into two sub-problems, namely the CP sub-problem and RRA sub-problem. Finally, we propose an iterative method and a Stackelberg game based method to solve them, and numerical results show that the proposed solution can significantly improve network profit compared to some existing schemes in NOMA-based F-RANs.