We identify and investigate the nature of the 20 brightest 250 mu m sources detected by the Balloon-borne Large Aperture Submillimetre Telescope (BLAST) within the central 150 arcmin(2) of the Great Observatories Origins Deep Survey (GOODS)-South field. Aided by the available deep VLA 1.4 GHz radio imaging, reaching S-1.4 similar or equal to 40 mu Jy (4 sigma), we have identified radio counterparts for 17/20 of the 250 mu m sources. The resulting enhanced positional accuracy of similar or equal to 1 arcsec has then allowed us to exploit the deep optical (Hubble Space Telescope), near-infrared (VLT) and mid-infrared (Spitzer) imaging of GOODS-South to establish secure galaxy counterparts for the 17 radio-identified sources, and plausible galaxy candidates for the three radio-unidentified sources. Confusion is a serious issue for this deep BLAST 250 mu m survey, due to the large size of the beam. Nevertheless, we argue that our chosen counterparts are significant, and often dominant contributors to the measured BLAST flux densities. For all of these 20 galaxies we have been able to determine spectroscopic (eight) or photometric (12) redshifts. The result is the first near-complete redshift distribution for a deep 250 mu m-selected galaxy sample. This reveals that 250 mu m surveys reaching detection limits of similar or equal to 40 mJy have a median redshift z similar or equal to 1, and contain not only low-redshift spirals/LIRGs, but also the extreme z similar or equal to 2 dust-enshrouded starburst galaxies previously discovered at sub-millimetre wavelengths. Inspection of the LABOCA 870 mu m imaging of GOODS-South yields detections of similar or equal to 1/3 of the proposed BLAST sources (all at z > 1.5), and reveals 250/870 mu m flux-density ratios consistent with a standard 40 K modified blackbody fit with a dust emissivity index beta = 1.5. Based on their Infrared Array Camera (IRAC) colours, we find that virtually all of the BLAST galaxy identifications appear better described as analogues of the M82 starburst galaxy, or Sc star-forming discs rather than highly obscured ULIRGs. This is perhaps as expected at low redshift, where the 250 mu m BLAST selection function is biased towards spectral energy distributions which peak longward of lambda(rest) = 100 mu m. However, it also appears largely true at z similar or equal to 2.