  Motivated by recent experimental progress on iron-based ladder compounds, we
study the doped two-orbital Hubbard model for the two-leg ladder BaFe$_2$S$_3$.
The model is constructed by using {\it ab initio} hopping parameters and the
ground state properties are investigated using the density matrix
renormalization group method. We show that the $(\pi,0)$ magnetic ordering at
half-filling, with ferromagnetic rungs and antiferromagnetic legs, becomes
incommensurate upon hole doping. Moreover, depending on the strength of the
Hubbard $U$ coupling, other magnetic patterns, such as $(0,\pi)$, are also
stabilized. We found that the binding energy for two holes becomes negative for
intermediate Hubbard interaction strength, indicating hole pairing. Due to the
crystal-field split among orbitals, the holes primarily reside in one orbital,
with the other one remaining half-filled. This resembles orbital selective Mott
states. The formation of tight hole pairs continues with increasing hole
density, as long as the magnetic order remains antiferromagnetic in one
direction. The study of pair-pair correlations indicates the dominance of the
intra-orbital spin-singlet channel, as opposed to other pairing channels.
Although in a range of hole doping pairing correlations decay slowly, our
results can also be interpreted as corresponding to a charge-density-wave made
of pairs, a precursor of eventual superconductivity after interladder couplings
are included. Such scenario of intertwined orders has been extensively
discussed before in the cuprates, and our results suggest a similar physics
could exist in ladder iron-based superconductors. Finally, we also show that a
robust Hund's coupling is needed for pairing to occur.
