We have investigated the crystal, electronic, and magnetic structures of Mn5O8 by means of state-of-the-art density functional theory calculations and neutron powder diffraction (NPD) measurements. This compound stabilizes in the monoclinic structure with space group C2/m where the Mn ions are in the distorted octahedral and trigonal prismatic coordinations with oxygen atoms. The calculated structural parameters based on total energy calculations are found to be in excellent agreement with low temperature NPD measurements when we accounted for the correct magnetic structure and Coulomb correlation effect in the computation. Using fully relativistic generalized-gradient approximation with Hubbard U (GGA+U) we found that the magnetic ordering in Mn5O8 is A-type antiferromagnetic and the direction of the easy axis is [1 0 0] in agreement with susceptibility and NPD measurements. However, the calculation without the inclusion of Hubbard U leads to ferrimagnetic half metal as a ground state contradictory to experimental findings, indicating the presence of a strong Coulomb correlation effect in this material. The GGA calculation without the Coulomb correction effect itself is sufficient to reproduce the experimentally observed magnetic moments in various Mn sites. We found that Mn in this material exhibits mixed valence behavior with 2+ and 4+ oxidation states reflecting different magnetic moments in the Mn sites. We explored the electronic band characteristics using total, site-, and orbital-projected density of states which emphasized the mixed-valent nature of Mn. A dominant Mn 3d character of the density of states at Fermi energy is the origin for the metallic behavior of Mn5O8. The bond strength analysis based on the crystal orbital Hamiltonian population between constituents indicates strong anisotropy in the bonding behavior which results from the layered nature of its crystal structure. Our bonding analysis shows that there is a noticeable covalent bond between Mn 3d-O 2p states which stabilizes the observed low symmetric structure. Our experimental findings and theoretical predictions suggest that Mn5O8 can be classified as a strongly correlated mixed valent antiferromagnetic metal.