Multi-principal element alloys and high-entropy alloys are currently extensively studied for their hydrogen storage abilities. In this context, they offer a wide variety of properties that can be precisely tuned by the proper design of, among others, the chemical composition, crystal structure, valence-electron number, or lattice distortion parameter. However, so far, the community's primary attention has focused on highly crystalline materials with body-centered-cubic solid solutions structure or intermetallic Laves phases (type C14 and C15), and very little attention has been paid to poorly crystalline or amorphous alloys. Here we present the results obtained for poorly crystalline TiVFeCuNb alloy synthesized by mechanical alloying. The alloy showed an even distribution of all constituent elements and thermal stability up to 450 °C. The high-pressure DSC experiments under optimized conditions (without activation, at 30 bar H2) demonstrated that the alloy absorbs hydrogen between 150 and 250 °C within a two-step reaction. The X-ray powder diffraction patterns proved that the poorly crystalline structure is kept after the hydrogenation process. The combined DSC/TG/MS studies revealed that the hydrogenated alloy releases up to 0.59 wt% of H2 in a multi-step desorption process (between 170 and 450 °C). The kinetic hydrogen absorption measurements showed that the studied alloy reached half its capacity within 20 s of the process.