Recently a new class of metal alloys, of single-phase multicomponent composition at roughly equal atomic concentrations ("equiatomic"), have been shown to exhibit promising mechanical, magnetic, and corrosion resistance properties, in particular, at high temperatures. These features make them potential candidates for components of next-generation nuclear reactors and other high-radiation environments that will involve high temperatures combined with corrosive environments and extreme radiation exposure. In spite of a wide range of recent studies of many important properties of these alloys, their radiation tolerance at high doses remains unexplored. In this work, a combination of experimental and modeling efforts reveals a substantial reduction of damage accumulation under prolonged irradiation in single-phase NiFe and NiCoCr alloys compared to elemental Ni. This effect is explained by reduced dislocation mobility, which leads to slower growth of large dislocation structures. Moreover, there is no observable phase separation, ordering, or amorphization, pointing to a high phase stability of this class of alloys.