Determining the spatial distribution and intrinsic physical properties of neutral hydrogen on cosmological scales is one of the key goals of next-generation radio surveys. We use the EAGLE galaxy formation simulations to assess the properties of damped Lyman α absorbers (DLAs) that are associated with galaxies and their underlying dark matter haloes between 0 ≤ z ≤ 2. We find that the covering fraction of DLAs increases at higher redshift; a significant fraction of neutral atomic hydrogen (H I) resides in the outskirts of galaxies with stellar mass ≥10^10 M☉; and the covering fraction of DLAs in the circumgalactic medium (CGM) is enhanced relative to that of the interstellar medium (ISM) with increasing halo mass. Moreover, we find that the mean density of the H I in galaxies increases with increasing stellar mass, while the DLAs in high- and low-halo mass systems have higher column densities than those in galaxies with intermediate halo masses (∼10^12 M☉ at z = 0). These high-impact CGM DLAs in high-stellar mass systems tend to be metal poor, likely tracing smooth accretion. Overall, our results point to the CGM playing an important role in DLA studies at high redshift (z ≥ 1). However, their properties are impacted both by numerical resolution and the detailed feedback prescriptions employed in cosmological simulations, particularly that of active galactic nuclei.