Compared with conventional regular hexagonal cellular models, random cellular network models resemble real cellular networks much more closely. However, most studies of random cellular networks are based on the Poisson point process and do not take into account the fact that adjacent base stations (BSs) should be separated with a minimum distance to avoid strong interference among each other. In this paper, based on the hard core point process (HCPP), we propose a multi-user multi-antenna random cellular network model with the aforementioned minimum distance constraint for adjacent BSs. Taking into account the effects of small scale fading and shadowing, interference and capacity models are derived for the multi-user multi-antenna HCPP random cellular networks. Furthermore, a spectrum efficiency model as well as an energy efficiency model is presented, based on which, the maximum achievable energy efficiency of the considered multi-user multiantenna HCPP random cellular networks is investigated. Simulation results demonstrate that the energy efficiency of conventional Poison point process (PPP) cellular networks is underestimated when the minimum distance between adjacent BSs is ignored.