A two-component density-functional theory is presented for electron-positron systems. The phase diagram of a two-component Fermi-Coulomb system is discussed, and explicit expressions are derived for exchange-correlation functionals for use in the local-density approximation. The scheme is then applied in a fully self-consistent calculation of electron and positron densities in atomic vacancies in metals, treated in the jellium model. Comparison with conventional calculations, which do not meet true electron-positron self-consistency, reveals considerable changes in the density distributions. However, we demonstrate that there are cancellation effects which render the corresponding changes in observable annihilation characteristics relatively small.