We predict a class of topological crystalline insulators in the antiperovskite material family with the chemical formula A(3)BX. Here the nontrivial topology arises from band inversion between two J = 3/2 quartets, which is described by a generalized Dirac equation for a "Dirac octet." Our work suggests that antiperovskites are a promising venue for exploring the cooperative interplay between band topology, crystal symmetry, and electron correlation.