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We report on the stellar content, half-light radii and star formation rates of a sample of 10 known high-redshift ($z\gtrsim 2$) galaxies selected on strong neutral hydrogen (HI) absorption (log(N(HI)/cm$^{-2})>19$) toward background quasars. We use observations from the {\it Hubble Space Telescope} (HST) Wide Field Camera 3 in three broad-band filters to study the spectral energy distribution(SED) of the galaxies. Using careful quasar point spread function subtraction, we study their galactic environments, and perform the first systematic morphological characterisation of such absorption-selected galaxies at high redshifts. Our analysis reveals complex, irregular hosts with multiple star-forming clumps. At a spatial sampling of 0.067 arcsec per pixel (corresponding to 0.55 kpc at the median redshift of our sample), 40% of our sample requires multiple S��rsic components for an accurate modelling of the observed light distributions. Placed on the mass-size relation and the `main sequence' of star-forming galaxies, we find that absorption-selected galaxies at high redshift extend known relations determined from deep luminosity-selected surveys to an order of magnitude lower stellar mass, with objects primarily composed of star-forming, late-type galaxies. We measure half-light radii in the range $r_{1/2} \sim$ 0.4 to 2.6 kpc based on the reddest band (F160W) to trace the oldest stellar populations, and stellar masses in the range $\log (\mathrm{M}_{\star}/\mathrm{M}_{\odot}) \sim$ 8 to 10 derived from fits to the broad-band SED. Spectroscopic and SED-based star formation rates are broadly consistent, and lie in the range log(SFR/M$_{\odot} $yr$^{-1}$) $\sim$0.0 to 1.7., 17 pages, Accepted for publication in MNRAS. This revision has minor text changes