Stable I-h-symmetric polygermane and polystannane nanostructures have been predicted on the basis of quantum chemical calculations. Structures, stabilities, and electronic properties of cages up to Ge500H500 and Sn500H500 were determined. The facets of the icosahedral cages resemble the monolayer sheets of experimentally known layered GeH. The stabilities and electronic properties of the cages converge toward the corresponding monolayer sheets. Due to their electronic properties, the polygermane and polystannane nanostructures may turn out to be useful in optical applications.