Two-dimensional metal–organic frameworks (2D-MOFs) with exotic electronic structures are drawing increasing attention. Here, using first-principles calculations, we demonstrate a spin-gapless MOF, namely, Mn₂C₆S₁₂, with the coexistence of a spin-polarized Dirac cone and parabolic degenerate points. The Curie temperature evaluated from Monte Carlo simulations implies Mn₂C₆S₁₂ possessing stable ferromagnetism at room temperature. Taking the spin–orbit coupling into account, the Dirac cone is gapped and the degenerate points are lifted, giving rise to multiple topologically nontrivial states with nonzero Chern number, which imply the possibility of Mn₂C₆S₁₂ to be a Chern insulator and a Chern half-metal. Our results offer versatile platforms for achieving spin filtering or a quantum anomalous Hall effect with promising application in spintronics devices.