We lay out the extension of range-separated density-functional theory to a four-component relativistic frame- work using a Dirac-Coulomb-Breit Hamiltonian in the no-pair approximation. This formalism combines a wave- function method for the long-range part of the electron-electron interaction with a density(-current) functional for the short-range part of the interaction. We construct for this formalism a short-range exchange local-density approximation based on calculations on a relativistic homogeneous electron gas with a modified Coulomb-Breit electron-electron interaction. More specifically, we provide the relativistic short-range Coulomb and Breit ex- change energies per particle of the relativistic homogeneous electron gas in the form of Pad ́e approximants which are systematically improvable to arbitrary accuracy. These quantities, as well as the associated effective Coulomb-Breit exchange hole, show the important impact of relativity on short-range exchange effects for high densities.