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Hydrogen is a key element in the changing energy sector and presents an accessible alternative to conventional fossil fuel sources. In this work, a system of ten high-entropy alloys was prepared based on the Hume-Rothery rules. One of the biggest advantages of these alloys is their storage capacity, which reaches the highest value among all known alloys intended for hydrogen storage. Alloys based on Al-Ti-Nb-Zr elements with different atomic fractions show interesting accumulation capabilities with fast absorption kinetics and low specific gravity. Each alloy in this study underwent high-pressure gravimetric absorption and desorption tests. The main goals of this work were to prepare alloys with the lowest-possible specific gravity and the highest-possible storage capacity. One alloy from our system shows storage capacity values similar to commercial alloys, without any rare-earth elements.