The neutron and gamma ray fluxes within the shielded high-temperature superconducting central columns of proposed spherical tokamak power plants have been studied using the MCNP Monte-Carlo code. The spatial, energy and angular variations of the fluxes over the shield and superconducting core are computed and used to specify experimental studies relevant to radiation damage and activation. The mean neutron and gamma fluxes, averaged over energy and angle, are shown to decay exponentially through the shield and then to remain roughly constant in the core region. The mean energy of neutrons is shown to decay more slowly than the neutron flux through the shield while the gamma energy is almost constant around 2 MeV. The differential neutron and gamma fluxes as a function of energy are examined. The neutron spectrum shows a fusion peak around 1 MeV changing at lower energies into an epithermal E−0.85 variation and at thermal energies to a Maxwellian distribution. The neutron and gamma energy spectra are defined for the outer surface of the superconducting core, relevant to damage studies. The inclusion of tungsten boride in the shield is shown to reduce energy deposition. A series of plasma scenarios with varying plasma major radii between 0.6 and 2.5 m was considered. Neutron and gamma fluxes are shown to decay exponentially with plasma radius, except at low shield thickness. Using the currently known experimental fluence limitations for high temperature superconductors, the continuous running time before the fluence limit is reached has been calculated to be days at 1.4 m major radius increasing to years at 2.2 m. This work helps validate the concept of the spherical tokamak route to fusion power by demonstrating that the neutron shielding required for long lifetime fusion power generation can be accommodated in a compact device.