We examine the properties of atomic hydrogen (H I) associated with galaxies in the Evolution and Assembly of GaLaxies and their Environments (EAGLE) simulations of galaxy formation. EAGLE's feedback parameters were calibrated to reproduce the stellar mass function and galaxy sizes at z=0.1, and we assess whether this calibration also yields realisticHI properties. We estimate the self-shielding density with a fitting function calibrated using radiation transport simulations, and correct for molecular hydrogen with empirical or theoretical relations. The 'standard-resolution' simulations systematically underestimate HI column densities, leading to an HI deficiency in low-mass (M* < 1010M⊙) galaxies and poor reproduction of the observed HI mass function. These shortcomings are largely absent from EAGLE simulations featuring a factor of 8 (2) bettermass (spatial) resolution, within which the HI mass of galaxies evolves more mildly from z = 1 to 0 than in the standard-resolution simulations. The largest volume simulation reproduces the observed clustering of HI systems, and its dependence on HI richness. At fixed M*, galaxies acquire more HI in simulations with stronger feedback, as they become associated with more massive haloes and higher infall rates. They acquire less HI in simulations with a greater star formation efficiency, since the star formation and feedback necessary to balance the infall rate is produced by smaller gas reservoirs. The simulations indicate that the HI of present-day galaxies was acquired primarily by the smooth accretion of ionized, intergalactic gas at z ≃ 1, which later self-shields, and that only a small fraction is contributed by the reincorporation of gas previously heated strongly by feedback. HI reservoirs are highly dynamic: over 40 per cent of HI associated with z = 0.1 galaxies is converted to stars or ejected by z = 0.