Due to the emergence of non-terrestrial platforms with extensive coverage, flexible deployment, and reconfigurable characteristics, the hybrid satellite-aerial-terrestrial networks (HSATNs) can accommodate a great variety of wireless access services in different applications. To effectively reduce the transmission latency and facilitate the frequent update of files with improved spectrum efficiency, we investigate the performance of cache-enabled HSATN, where the user retrieves the required content files from the cache-enabled aerial relay or the satellite with the non-orthogonal multiple access (NOMA) scheme. If the required content files of the user are cached in the aerial relay, the cache-enabled relay would serve directly. Otherwise, the user would retrieve the content file from the satellite system, where the satellite system seeks opportunities for proactive content pushing to the relay during the user content delivery phase. Specifically, taking into account the uncertainty of the number and location of aerial relays, along with the channel fading of terrestrial users, the outage probability and hit probability of the considered network are, respectively, derived based on stochastic geometry. Numerical results unveil the effectiveness of the cache-enabled HSATN with the NOMA scheme and proclaim the influence of key factors on the system performance. The realistic, tractable, and expandable framework, as well as associated methodology, provide both useful guidance and a solid foundation for evolved networks with advanced configurations in the performance of cache-enabled HSATN.