  Motivated by the discovery of superconductivity in the two-leg, quasi-one
dimensional ladder compound, BaFe$_2$S$_3$ we present a renormalization group
study of electrons moving on a two leg, two orbital ladder, subjected to
Hubbard repulsion $U$ and Hund's coupling $J$. In our calculations, we adopt
tightbinding parameters obtained from ab-initio studies on this material. At
incommensurate filling, the long wavelength analysis displays four phases as a
function of $0 \leq J/U <1$. We show that a fully gapped superconductor is
stabilized at sufficiently large Hund's coupling, the relative phases at the
three Fermi points are "+,-,-". By contrast, when the system is tuned to half
filling, Umklapp scattering gives rise to Mott insulating phases. We discuss
the general implications of our study for the broad class of iron-based
superconductors.
