We use state-of-the-art cosmological zoom simulations to explore the distribution of neutral gas in and around galaxies that gives rise to high column density \ion Hi Ly- \alpha absorption ( formally , sub-DLAs and DLAs ) in the spectra of background quasars . Previous cosmological hydrodynamic simulations under-predict the mean projected separations ( b ) of these absorbers relative to the host , and invoke selection effects to bridge the gap with observations . On the other hand , single lines of sight ( LOS ) in absorption can not uniquely constrain the galactic origin . Our simulations match all observational data , with DLA and sub-DLA LOS existing over the entire probed parameter space ( -4 \lesssim [ M/H ] \lesssim 0.5 , b < 50 kpc ) at all redshifts ( z \sim 0.4 - 3.0 ) . We demonstrate how the existence of DLA LOS at b \gtrsim 20 - 30 kpc from a massive host galaxy require high numerical resolution , and that these LOS are associated with dwarf satellites in the main halo , stripped metal-rich gas and outflows . Separating the galaxy into interstellar ( ‘ ‘ \ion Hi disc '' ) and circumgalactic ( ‘ ‘ halo '' ) components , we find that both components significantly contribute to damped \ion Hi absorption LOS . Above the sub-DLA ( DLA ) limits , the disc and halo contribute with \sim 60 ( 80 ) and \sim 40 ( 20 ) per cent , respectively . Our simulations confirm analytical model-predictions of the DLA-distribution at z \lesssim 1 . At high redshift ( z \sim 2 - 3 ) sub-DLA and DLAs occupy similar spatial scales , but on average separate by a factor of two by z \sim 0.5 . On whether sub-DLA and DLA LOS sample different stellar-mass galaxies , such a correlation can be driven by a differential covering-fraction of sub-DLA to DLA LOS with stellar mass . This preferentially selects sub-DLA LOS in more massive galaxies in the low- z universe .