Chloride perovskites are semiconductors with a near-ultraviolet bandgap that are promising for applications in optoelectronics and photonics. One of the most studied representatives of this family is the methylammonium lead chloride MAPbCl3 (MA+ = CH3NH3+). Low-temperature luminescence spectroscopy of this material demonstrates a complex emission structure. In this work, we have studied the cathodoluminescecne of the MAPbCl3 halide perovskite single crystal at 70 K. Excitation by an electron beam was used to localize different emitters: excitons, defect-related states, and inclusions, previously assigned to the material itself. Exciton luminescence is observed from an undamaged sample, while the defect band is emitted from regions with dislocations, growth defects, and crystal damage. Defect formation under electron beam irradiation was studied. It was found that MAPbCl3 is resistant to irradiation, which supports the defect tolerance of halide perovskites and paves the way for their electron-beam modification for applications.