Metal hydrides (MHx) provide a promising solution for the requirement to store large amounts of hydrogen in a future hydrogen-based energy system. This requires the design of alloys which allow for a very high H/M ratio. Transition metal hydrides typically have a maximum H/M ratio of 2 and higher ratios can only be obtained in alloys based on rare-earth elements. In this study we demonstrate, for the first time to the best of our knowledge, that a high entropy alloy of TiVZrNbHf can absorb much higher amounts of hydrogen than its constituents and reach an H/M ratio of 2.5. We propose that the large hydrogen-storage capacity is due to the lattice strain in the alloy that makes it favourable to absorb hydrogen in both tetrahedral and octahedral interstitial sites. This observation suggests that high entropy alloys have future potential for use as hydrogen storage materials.