Low-permeability, dense sandstone formations are ideal for underground gas storage in compressed air energy storage. To examine the gas permeability of fine sandstone under high pressure, a self-developed triaxial high-pressure gas permeameter was used for experiments under varying confining and axial pressures. The study identified unsteady seepage variations and stress dependence in porous media represented by fine sandstone and developed a gas leakage model. Results indicate that the seepage process includes rapid, slow, and cessation stages. Fine sandstone shows excellent storage capacity, with a daily leakage rate of 0.0463%, below required thresholds. This study guides high-pressure gas permeability theories and modeling.