H-net, a two-dimensional carbon allotrope incorporating distorted squares C4, hexagons C6, and octagons C8, is proposed using first principle calculations. Different from the previous planar graphene and other theoretical carbon sheets, H-net is a two-atom-thick polymorph with identical C6+C4+C6 components cross facing and covalently buckled to feature a handshake-like model. The feasible existence of H-net is evident from its dynamic stability as confirmed by phonon-mode analysis and energy superiority. H-net is energetically more favorable than synthesized graphdiyne and theoretical graphyne, BPC, S-graphene, polycyclic net, α-squarographite, and lithographite. Therefore, we address a possible synthetic route from graphene nanoribbon. Electronic band structure calculations indicate that H-net is a semiconductor with an indirect band gap of 0.88 eV, whereas graphene and many two-dimensional carbon sheets are metallic. We also explore the electronic structure properties of the one-dimensional derivative nanoribbons of H-net. The narrowest H-net nanoribbon exhibits metallic behavior, and others are semiconducting with growing band gaps as nanoribbon widths. H-net and its tailored nanoribbons are expected to possess more properties than graphene because of their exceptional crystal structure and different energy band.