  We study a versatile structurally favorable periodic $sp^2$-bonded carbon
atomic planar sheet with $C_{4v}$ symmetry by means of the first-principles
calculations. This carbon allotrope is composed of carbon octagons and squares
with two bond lengths and is thus dubbed as octagraphene. It is a semimetal
with the Fermi surface consisting of one hole and one electron pocket, whose
low-energy physics can be well described by a tight-binding model of
$\pi$-electrons. Its Young's modulus, breaking strength and Poisson's ratio are
obtained to be 306 $N/m$, 34.4 $N/m$ and 0.13, respectively, which are close to
those of graphene. The novel sawtooth and armchair carbon nanotubes as well as
unconventional fullerenes can also be constructed from octagraphene. It is
found that the Ti-absorbed octagraphene can be allowed for hydrogen storage
with capacity around 7.76 wt%.
