In the field of deep geothermal energy extraction and material storage (such as carbon storage, hydrogen storage, and waste storage), permeability is one of the crucial parameters for evaluating the sealing performance of reservoirs or barrier systems. Different testing methods, fluid media, and testing conditions can all influence the results of permeability tests. This paper first introduces four steady-state methods and transient methods, providing the derivation process of the steady-state method based on pressure decay, supplemented by self-correction of pressure fluctuations caused by temperature fluctuations. Taking engineered barriers of hazardous waste disposal facilities as the research object, permeability tests and calculations for different media (water/air) were conducted. The results indicate that the gas permeability of compacted bentonite samples after saturation (10–21 m2) is an order of magnitude lower than the water permeability (10–20 m2). However, when the gas pressure gradually increases, gas breakthrough occurs, leading to an increase in gas permeability to 10–18 m2. This suggests that gas breakthrough has a significant impact on the formation of gas migration pathways. The permeability calculation method proposed in this paper can provide some guidance and reference for the selection of testing methods for subsequent low-permeability media.