What materials can block ionising radiation?

The type of material depends on the type of ionising radiation in question.

Alpha Particles - single sheet of paper, the outer dead layer of skin of your body, clothing

Beta Particles - depends in energy of beta particle, but 12mm of a low density material like perspex or plastic will stop all betas

Gamma Ray / X-ray - shielding performance depends on energy of the radiation for a given thickness and density. Higher density shielding such as lead / tungsten will provide the best shielding performance for the least thickness. If space is of little concern high density concrete, normal concrete or even water can be used (but these shields will be thicker).