Spin transfer torque magnetic random access memory (STT-MRAM) is a promising candidate for next generation memory as it is non-volatile, fast, and has unlimited endurance. Another important aspect of STT-MRAM is that its core component, the nanoscale magnetic tunneling junction (MTJ), is thought to be radiation hard, making it attractive for space and nuclear technology applications. However, studies on the effects of ionizing radiation on the STT-MRAM writing process are lacking for MTJs with perpendicular magnetic anisotropy (pMTJs) required for scalable applications. Particularly, the question of the impact of extreme total ionizing dose on perpendicular magnetic anisotropy, which plays a crucial role on thermal stability and critical writing current, remains open. Here we report measurements of the impact of high doses of gamma and neutron radiation on nanoscale pMTJs used in STT-MRAM. We characterize the tunneling magnetoresistance, the magnetic field switching, and the current-induced switching before and after irradiation. Our results demonstrate that all these key properties of nanoscale MTJs relevant to STT-MRAM applications are robust against ionizing radiation. Additionally, we perform experiments on thermally driven stochastic switching in the gamma ray environment. These results indicate that nanoscale MTJs are promising building blocks for radiation-hard non-von Neumann computing.