We study the morphology and size of the luminous and massive galaxies at 0.3 < z < 0.7 targeted in the Baryon Oscillation Spectroscopic Survey (BOSS) using publicly available Hubble Space Telescope (HST) imaging, and catalogues, from the COSMic Origins Survey (COSMOS). Our sample (240 objects) provides a unique opportunity to check the visual morphology of these galaxies which were targeted based solely on stellar population modelling. We find that the majority of BOSS galaxies (74 ± 6 per cent) possess an early-type morphology (elliptical or lenticular), while the remainder have a late-type (spiral disc) morphology. This is as expected from the goals of the BOSS target selection which aimed to predominantly select slowly evolving galaxies, for use as cosmological probes, while still obtaining a fair fraction of actively star-forming galaxies for galaxy evolution studies. We show that a colour cut of (g−i) > 2.35 is able to select a sub-sample of BOSS galaxies with ≥90 per cent early-type morphology and thus more comparable to the earlier Luminous Red Galaxy (LRG) samples of Sloan Digital Sky Survey (SDSS)-I/II. The remaining ≃10 per cent of galaxies above this (g−i) cut have a late-type morphology and may be analogous to the ‘passive spirals’ found at lower redshift. We find that 23 ± 4 per cent of the early-type BOSS galaxies are unresolved multiple systems in the SDSS imaging. We estimate that at least 50 per cent of these multiples are likely real associations and not projection effects and may represent a significant ‘dry merger’ fraction. We study the SDSS pipeline sizes of BOSS galaxies which we find to be systematically larger (by 40 per cent) than those measured from HST images, and provide a statistical correction for the difference. These details of the BOSS galaxies will help users of the BOSS data fine-tune their selection criteria, dependent on their science applications. For example, the main goal of BOSS is to measure the cosmic distance scale and expansion rate of the Universe to per cent level precision – a point where systematic effects due to the details of target selection may become important.