Several select geopolymer, polymer, and composite materials are considered as potential candidates for the inner shielding in containers used for storage and disposal of low- and intermediate-level radioactive waste, from the perspective of radiation effects. The suitability of the probed materials is examined through Monte Carlo simulations, which yield absorbed dose values in container inner shieldings of various compositions and dimensions. The radioactive waste considered in the simulation models contains 60Co or 137Cs and is placed inside standard 216.5-L (55-gal) drums, in either compacted or solidified form. The influence of container stacking, in either a storage or a disposal environment, on the dose in the shielding is also taken into account. The simulation results are used for calculating the dose-dependent overpressure within the container caused by the gas generated in the inner shielding through radiolysis. Two types of waste activity limits are determined for each of the researched shielding materials: one below which the overpressure decreases after the initial heat-induced jump and another that results in an overpressure that stays just below the maximum tolerable value. Dose-dependent changes of the polymer and composite shielding materials’ molecular weights are also calculated. The obtained results show that with regard to the radiation effects caused by the investigated sources, the examined materials are compatible with the proposed use as inner shielding in radioactive waste containers.