To provide quantitative data for the development of underground hydrogen storage in porous sedimentary rocks, capillary pressures and relative permeabilities have been measured for the hydrogen–water system. The tests have been performed on a Triassic sandstone. Two potential underground hydrogen storage conditions (“shallower”: 55 bar, 20 ∘C and “deeper”: 100 bar, 45 ∘C) have been investigated. Capillary pressure curves have been measured following a modified semi-dynamic technique. The data have been combined with mercury injection capillary pressure measurements to derive a model for capillary pressure valid over almost the entire water saturation range. Interfacial tensions and contact angles for the hydrogen–water system have been also derived. Relative permeability curves measured with the steady-state technique yield low values for minimum water saturations of ∼ 40%. When combined with the capillary pressure data, the relative permeability of hydrogen in sandstone can be evaluated for almost the total range of water saturation. Capillary numbers calculated for our relative permeability experiments indicate a capillary-limited flow regime for the hydrogen–water system. Despite the two differing sets of conditions investigated and this flow regime, the relative permeability curves stay very close from each other, an effect attributed to the almost constant viscosity of hydrogen under our pressure and temperature conditions. This is in contrast with other fluid pairs (e.g., CO2–water system) where capillary numbers can strongly vary with pressure and temperature. Similarly, capillary pressure data vary little between the experimental conditions. The interpretation of the results would suggest that the relative permeability and capillary pressure results from this study are applicable to a wide range of pressure and temperature conditions.