Motivated by the recent successful formation of the MoSi2N4 monolayer [Hong et al., Sci. 369, 670 (2020)], the structural, electronic and magnetic properties of MoSi2N4 nanoribbons (NRs) is investigated for the first time . The band structure calculations showed spin-polarization in zigzag edges and a non-magnetic semiconducting character in armchair edges. For armchair-edges, we identify an indirect to direct band gap shift compared to the MoSi2N4 monolayer, and its energy gap increases with increasing NR width. Anisotropic electrical and magnetic behavior is observed via band structure calculations in the zigzag and armchair edges, where, surprisingly, for the one type of zigzag-edges configuration, we identify a Dirac-semimetal character. The appearance of magnetism and Dirac-semimetal in MoSi2N4 ribbon can give rise to novel physical properties, which could be useful in applications for next-generation electronic devices.