Most materials with high thermal conductivity belong to one of two classes: metals, where heat is carried predominantly by electrons, and insulators, where heat transport is dominated by the phonon contribution. Materials that combine substantial electronic thermal conductivity and lattice thermal conductivity are rare, because the mechanisms that favor electron transport typically suppress phonon transport, and vice versa. Here, we report the theoretical prediction and experimental realization of such a material, metallic $\epsilon$-TaN. Our calculations predict a total thermal conductivity at room-temperature of 273$$5Wm$^{-1}$K${-1}$ in single crystals and 145$$5Wm$^{-1}$K${-1}$ in polycrystals with 0.5$\mu$m grains, with an unusually large lattice contribution (79%) for a metal. The latter value is in agreement with our local transient thermoreflectance measurements on polycrystalline samples yielding $$130Wm$^{-1}$K${-1}$. We show that the balanced electronic and lattice thermal conductivities of $\epsilon$-TaN originate from a combination of large Fermi velocity and small Fermi density of states on the electron side, and large speed of sound and wide phonon gap on the lattice side.