We provide equation-of-state data for hard, uniaxial hyperellipsoids of revolution (hyperspheroids) in the four-dimensional Euclidean space for various prolate and oblate aspect ratios using virial coefficients up to the eighth order. Using analytically known second virial coefficients from Brunn–Minkowski theory as a reference, higher-order virial coefficients are calculated by means of Mayer-sampling Monte Carlo integration optimized for hard body systems. We compare results of hyperspheroids with spheroids and ellipses as their sections in two- and three-dimensional subspaces. To analyze the influence of the detailed particle shape in addition to the aspect ratio, we compare virial coefficients of hyperspheroids and hyperspherocylinders providing so far unknown seventh- and eighth-order virial coefficients of hyperspherocylinders. We compare equation-of-state data of anisotropic solids in R2, R3, and R4, demonstrating faster convergence of the virial series with increasing dimension of space.