High-entropy alloys (HEAs) are a new class of metallic alloys without a principal component. These materials are attractive because of their unique structures and properties, including mechanical ones. Some of HEAs based on refractory metals are considered as advanced high-temperature materials. However, the development of new applicable compositions is complicated by poor understanding of relationships between the chemical and phase composition of HEAs and their effect on mechanical performance. Here we report the results of comprehensive studies of Al-Cr-Nb-Ti-V-Zr high-entropy alloys, particularly AlCr x NbTiVZr y (х = 0.00, 0.25, 0.50, 1.00, 1.50 at y = 0 and y = 0.0, 0.25, 0.50, 1.00, 1.50 at x = 0). The strength and ductility of such alloys are shown to have a complex relationship with the composition and order of the parent B2 phase, and the amount and nature of secondary phases (C14 Laves phase and zirconium aluminides). Some of the investigated alloys demonstrate high specific strength at T ≤ 800°C. AlNbTiVZr 0.25 alloy also has a stable microstructure and higher creep resistance at 600°C compared to existing creep-resistant alloys. Based on calculations of equilibrium and nonequilibrium phase diagrams by the CALPHAD method, we propose new Al-Cr-Nb-Ti-Zr alloy compositions with unique lamellar eutectic structure composed of the B2 and C14 Laves phases and with enhanced strength properties. The structure/property relationships and high-temperature deformation mechanisms of refractory HEAs with eutectic microstructure are analyzed.