This paper aims to evaluate the low-velocity impact responses and mechanical properties of balsa-wood and urethane-foam core materials and their sandwich panels, which are applied as the impact limiter of a nuclear spent fuel shipping cask. For the urethane-foam core, which is isotropic, tensile, compressive, and shear mechanical tests were conducted. For the balsa-wood core, which is orthotropic and shows different material properties in different orthogonal directions, nine mechanical properties were determined. The impact test specimens for the core material and their sandwich panel were subjected to low-velocity impact loads using an instrumented testing machine at impact energy levels of 1, 3, and 5 J. The experimental results showed that both the urethane-foam and the balsa-wood core except in the growth direction (z-direction) had a similar impact response for the energy absorbing capacity, contact force, and indentation. Furthermore, it was found that the urethane-foam core was suitable as an impact limiter material owing to its resistance to fire and low cost, and the balsa-wood core could also be strongly considered as an impact limiter material for a lightweight nuclear spent fuel shipping cask.