Electronic structure calculations were performed on the compound BaCoO3. A-type antiferromagnetism is predicted as a ground state, with a very small Néel temperature, consistent with experiment. The insulating behavior of the compound needs symmetry breaking and occurs via an in-chain orbital ordering. The origin of this orbital ordering is explained. A large value of the orbital angular momentum is estimated, with the z axis being the easy direction for the magnetization. A large, Ising-type, magnetocrystalline anisotropy is calculated for this quasi-one-dimensional compound, consistent with the collinear magnetic structure predicted.