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Novel multicomponent Nb–Ni–Ti–Zr–Co alloys with 20–55 at.% Nb were synthesized from metal powders by arc melting. The resulting alloys consist primarily of Nb-rich and eutectic body-centered (BCC) phases. The content of the eutectic BCC phase is highest for an equimolar composition, while the content of the Nb-rich BCC phase increases with Nb content in the alloy. The content of secondary phases is the highest for the alloy with 32 at.% Nb. According to ab initio calculations, hydrogen occupies tetrahedral interstitial sites in the Nb-rich phase and octahedral sites in the eutectic BCC phase. For different Nb concentrations, hydrogen-binding energies were calculated. An increase in the Nb-rich phase leads to softening of multicomponent alloys. The alloys with 20 and 32 at.% Nb demonstrate high hydrogen permeability (1.05 and 0.96 × 10−8 molH2m−1s−1Pa−0.5, respectively) at 400 °C, making them promising for hydrogen purification membrane application. Multicomponent alloys with a high Nb content (55 at.%) have low resistance to hydrogen embrittlement.