Carbon-based molecular junctions consisting of aromatic oligomers between conducting sp hybridized carbon electrodes exhibit structure-dependent current densities (J) when the molecular layer thickness (d) exceeds ∼5 nm. All four of the molecular structures examined exhibit an unusual, nonlinear ln J vs bias voltage (V) dependence which is not expected for conventional coherent tunneling or activated hopping mechanisms. All molecules exhibit a weak temperature dependence, with J increasing typically by a factor of 2 over the range of 200-440 K. Fluorene and anthraquinone show linear plots of ln J vs d with nearly identical J values for the range d = 3-10 nm, despite significant differences in their free-molecule orbital energy levels. The observed current densities for anthraquinone, fluorene, nitroazobenzene, and bis-thienyl benzene for d = 7-10 nm show no correlation with occupied (HOMO) or unoccupied (LUMO) molecular orbital energies, contrary to expectations for transport mechanisms based on the offset between orbital energies and the electrode Fermi level. UV-vis absorption spectroscopy of molecular layers bonded to carbon electrodes revealed internal energy levels of the chemisorbed films and also indicated limited delocalization in the film interior. The observed current densities correlate well with the observed UV-vis absorption maxima for the molecular layers, implying a transport mechanism determined by the HOMO-LUMO energy gap. We conclude that transport in carbon-based aromatic molecular junctions is consistent with multistep tunneling through a barrier defined by the HOMO-LUMO gap, and not by charge transport at the electrode interfaces. In effect, interfacial "injection" at the molecule/electrode interfaces is not rate limiting due to relatively strong electronic coupling, and transport is controlled by the "bulk" properties of the molecular layer interior.