Recently, a novel two-dimensional (2D) Dirac semimetal BeN4 have been successfully synthesized (Phys. Rev. Lett. 126, 175,501, 2021). However, the zero bandgap of monolayer BeN4 prevents it from performing as a semiconductor in next-generation optoelectronics and electronics. Here we employ first-principles calculations to show that cutting BeN4 monolayer along zigzag direction can open moderate band gaps. The zigzag BeN4 nanoribbons with Be termination are nonmagnetic (NM) semiconductors with a moderate direct band gap and good optical absorption. The nanoribbons with edges occupied by Be and N atoms are nonmagnetic semiconductors or ferromagnetic (FM)/NM metals, and the electronic properties are closely related to the type of boundary N atoms. The configuration with both edges occupied by the pentagonal N atoms is an antiferromagnetic (AFM) semiconductor, while the system with both edges situated by hexagonal N atoms is a NM direct band gap semiconductor. If one edge is terminated by hexagonal N atoms and the other one is pentagonal N atoms, the nanoribbon will behave as a FM semiconductor. We also explore the electronic properties of zigzag BeN4 nanoribbons with hydrogen passivation. These excellent properties endow zigzag BeN4 nanoribbons with enormous potential for applications in optoelectronic and spintronic devices.