A theoretical study of the optical properties of a relativistic Fermi gas at a finite temperature is performed. The results are obtained from Maxwell’s equations, assuming bianisotropic constitutive relations for the relativistic gas. The longitudinal and transverse spatial directions are defined according to the properties of the electric permittivity tensor. The electromagnetic modes and the refractive index of the relativistic gas are calculated along such directions. The longitudinal and transverse group velocities corresponding to the respective propagation modes are also obtained. No typical characteristics of metamaterials are observed, neither in the refractive index nor in the group velocities corresponding to the transverse modes. However, in the case of longitudinal modes, negative values of the group velocity are observed in a region of the reciprocal space where particle–antiparticle excitations can occur.