The 2D naphthylene-β structure is a theoretically proposed sp2 nanocarbon allotrope based on the assembly of naphthalene-based molecular building blocks, which features metallic properties. We report that 2D naphthylene-β structures host a spin-polarized configuration which turns the system into a semiconductor. We analyze this electronic state in terms of the bipartition of the lattice. In addition, we study the electronic properties of nanotubes obtained from the rolling up of 2D naphthylene-β. We show that they inherit the properties of the parent 2D nanostructure, such as the emergence of spin-polarized configurations. We further rationalize the results in terms of a zone-folding scheme. We also show that the electronic properties can be modulated using an external transverse electric field, including a semiconducting-to-metallic transition for sufficiently large field strength.