We here identify by ab initio calculations two distinct three-dimensional three-connected (3D3C) chiral framework structures of carbon in and I41/amd symmetry, respectively, which comprise 3-fold and 4-fold helical chains with complementary chirality. The helical carbon chains are connected by an ethene-type planar π-conjugation, and the resulting structures contain a network of sp2 carbon bonds with one-third being double bonds between the chains and two-thirds single bonds along the chains. Phonon and electronic band structure calculations show that these chiral carbene structures are dynamically stable and exhibit a large band gap (2.4 ~ 2.9 eV). This semiconducting nature reflects a key distinction from previously proposed metallic isomers of helical or zigzag carbon chains with twisted π states that are dynamically unstable. The present results solve the long-sought 3D3C all-sp2 carbon structures and may help design other covalent bonding networks.