To effectively reduce the transmission latency and facilitate the frequent update of files with improved spectrum efficiency, we investigate the performance of satellite-aerial-terrestrial networks (HSATNs), where the user would retrieve the content file from the satellite system, and the satellite system seeks opportunities for proactive content pushing to the relay during the user content delivery phase with the non-orthogonal multiple access (NOMA) scheme. Specifically, taking into account the uncertainty of the number and location of aerial relays, along with the uncertainty of locations and 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.