A gas storage apparatus includes a foam body having a negative Poisson's ratio, and a gas impregnated in closed cells of the foam body, the gas having a pressure above atmospheric pressure. A method of making a gas storage apparatus includes providing a first body comprising a precursor material, and, while the first body is in an atmosphere including a gas, applying a stimulus to the precursor material, the stimulus causing the precursor material to form a closed-cell porous foam structure having a negative Poisson's ratio. Cells of the closed-cell porous foam structure encapsulate the gas at a pressure above atmospheric pressure.