We have studied the electronic properties and relative stability of the biphenylene sheet composed of alternating eight-, six- and four-carbon rings and its one-dimensional derivatives including ribbons and tubes of different widths and morphologies by means of density functional theory calculations. The two-dimensional sheet presents a metallic character that is also present in the planar strips with zigzag-type edges. Armchair-edged strips develop a band gap that decreases monotonically with the ribbon width. The narrowest armchair strip considered here (0.62 nm wide) presents a large band gap of 1.71 eV, while the 2.14 nm wide armchair strip exhibits a band gap of 0.08 eV. We have also found that tubes made by rolling these ribbons in a seamlessly manner are all metallic, independent of their chirality. However, while the calculated energy landscape suggests that planar strips present a relative stability comparable to that of C(60), in the tubular form, they present a more pronounced metastable nature with a Gibbs free energy of at least 0.2 eV per carbon higher than in C(60).