A　novel　two-dimensional　carbon　allotrope,　rectangular　graphyne　(R-graphyne)　with　tetra-rings　and　acetylenic　linkages,　is　proposed　by　first-principles　calculations.　Although　the　bulk　R-graphyne　exhibits　metallic　property,　the　nanoribbons　of　R-graphyne　show　distinct　electronic　structures　from　the　bulk.　The　most　intriguing　feature　is　that　band　gaps　of　R-graphyne　nanoribbons　oscillate　between　semiconductor　and　metallic　states　as　a　function　of　width.　Particularly,　the　zigzag　edge　nanoribbons　with　half-integer　repeating　unit　cell　exhibit　unexpected　Dirac-like　fermions　in　the　band　structures.　The　Dirac-like　fermions　of　the　R-graphyne　nanoribbons　originate　from　the　central　symmetry　and　two　sub-lattices.　The　extraordinary　properties　of　R-graphyne　nanoribbons　greatly　expand　our　understanding　of　the　origin　of　Dirac-like　points.　Such　findings　uncover　a　novel　fascinating　property　of　nanoribbons,　which　may　have　broad　potential　applications　for　carbon-based　nanoscale　electronic　devices.