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The effect of epitaxial strain on structural, magnetic and electronic properties of BaFeO perovskite oxide are investigated from first principles calculations, using the density functional theory (DFT) plus the Hubbard approach (DFT+U) within the generalized gradient approximation. Hybrid functional calculations, based on mixed exact Hartree-Fock and DFT exchange energy functionals, are also performed. For the ground state calculations, the DFT+U is found more suitable to describe the half metallic and ferromagnetic state of cubic BaFeO. Jahn-Teller distortions, oxygen octahedra rotations and charge orderings in BaFeO under biaxial strain are explored. The obtained results reveal that structural changes associated with Jahn-Teller distortions are induced under tensile biaxial strain while the oxygen octahedra rotations and breathing are unusually not observed. Then, the strained BaFeO is considered as a particular Jahn-Teller distorted perovskite with exceptional properties when compared to CaFeO and SrFeO. These findings lead to a strain engineering of the JT distortions in BaFeO, and thus for high fundamental and technological interests.