We use cosmological hydrodynamic zoom simulations to study the neutral gas distribution in and around galaxies that gives rise to high column density H I Ly α absorption (damped Ly α systems (DLAs) and sub-DLAs) in background quasar spectra. Such simulations often sacrifice numerical resolution for volume that affects the lower density galaxy halo gas, and simulations have difficulties reproducing the span of projected separations (b) between absorbing clouds and their hosts. Our simulations produce (sub-)DLAs over the entire probed parameter space (b ≲ 50 kpc and metallicity −4 ≲[M/H]≲ 0.5) at all redshifts (z ∼ 0.4 − 3.0), enclosing spectroscopically confirmed absorber-galaxy pairs. Recovering (sub-)DLAs at b ≳ 20 − 30 kpc from a massive host galaxy requires high numerical resolution and efficient feedback, and we show that these lines-of-sight are associated with dwarf satellites in the main halo, stripped metal-rich gas, and outflows. H I disc- and halo gas significantly contributes to (sub-)DLAs around galaxies. At large redshifts the halo plays an increasingly important role, while at 0.4 < z < 1 the disc and halo contribute with ∼60(80) and ∼40(20) per cent to column densities above the sub-DLA(DLA) lower limits. The distribution of b for sub-DLAs and DLAs overlap at z ∼ 2 − 3, but evolves so that sub-DLAs on average are located at twice larger b by z ∼ 0.5. A weak correlation suggests that sub-DLA covering fractions increase with stellar mass more rapidly than those of DLA. This can explain why sub-DLAs are preferentially selected in more massive galaxies in the low-z Universe.