We have performed extensive density matrix renormalization group (DMRG) studies of the Hubbard model on a honeycomb ladder. The band structure (with Hubbard U = 0) exhibits an unusual quadratic band touching at half-filling, which is associated with a quantum Lifshitz transition from a band insulator to a metal. For one electron per site, nonzero U drives the system into an insulating state in which there is no pair-binding between added electrons; this implies that superconductivity driven directly by the repulsive electron-electron interactions is unlikely in the regime of small doping, x << 1. However, the divergent density of states as x -> 0, the large values of the phonon frequencies, and an unusual correlation induced enhancement of the electron-phonon coupling imply that lightly doped polyacenes, which approximately realize this structure, are good candidates for high-temperature electron-phonon driven superconductivity.