The sources discovered in deep hard X-ray surveys with 2-8 keV fluxes of S (2-8) similar to 10(-14) erg cm(-2) s(-1) make up the bulk of the X-ray background at these energies. Here we present detailed multiwavelength observations of three such sources from the European Large-Area ISO Survey Deep X-ray Survey. The observations include sensitive near-infrared spectroscopy with the Subaru Telescope and X-ray spectral information from the Chandra X-ray Observatory . The sources observed all have optical-to-near-infrared (near-IR) colours redder than an unobscured quasar and comprise a reddened quasar, a radio galaxy and an optically obscured active galactic nucleus (AGN). The reddened quasar is at a redshift z = 2.61 and shows a very large X-ray absorbing column of N (H) approximate to 3 x 10(23) cm(-2). This contrasts with the relatively small amount of dust reddening, implying a gas-to-dust ratio along the line of sight 100 times greater than that of the Milky Way. The radio galaxy at z = 1.57 shows only narrow emission lines, but has a surprisingly soft X-ray spectrum. The softness of this spectrum indicates either an unusually low gas-to-dust ratio for the absorbing medium or X-ray emission related to the young radio source. The host galaxy is extremely red (R -K = 6.4) and its optical/near-IR spectrum is best fit by a strongly reddened (A(V) approximate to 2) starburst. The third X-ray source discussed is also extremely red (R -K = 6.1) and lies in a close grouping of three other R -K > 6 galaxies. No emission or absorption lines were detected from this object, but its redshift (and that of one of the nearby galaxies) is constrained by spectral energy distribution fitting to be just greater than z = 1. The extremely red colours of these two galaxies can be accounted for by old stellar populations. These observations illustrate the diverse properties of hard X-ray-selected AGN at high redshift in terms of obscuration at optical and X-ray wavelengths and the evolutionary states of their host galaxies.