A rectangular graphyne sheet is composed of units similar to phenyl rings that are linked by acetylenic chains, as in hexagonal γ-graphyne. This system is organized over a rectangular lattice similar to that of the recently synthesized biphenylene network. We investigate the stability of this layered material from different perspectives and study its electronic structure. Rectangular graphyne is a semiconducting system in its pristine form. It features a pair of highly localized states. These characteristics are correlated with the structural anisotropy of the system, since its frontier states behave like quasi–1D states embedded in the 2D lattice. We further consider modified systems in which longer acetylenic links are introduced. We discuss how a strategic choice of the position of these longer bridges leads to specific changes of the electronic structure of the rectangular graphyne sheet.