Aims: We present optical photometry and spectroscopy of the afterglow and host galaxy of gamma-ray burst GRB 040924. This GRB had a rather short duration of T90 ~2.4 s, and a well sampled optical afterglow light curve. We aim to use this dataset to find further evidence that this burst is consistent with a massive star core-collapse progenitor. Methods: We combine the afterglow data reported here with those from the literature and compare the host properties with survey data. Results: We find that the global behaviour of the optical afterglow is well fit by a broken power-law, with a break at ~0.03 days. We determine the redshift z = 0.858 ± 0.001 from the detected emission lines in our spectrum. Using the spectrum and photometry we derive global properties of the host, showing it to have similar properties to other long GRB hosts. We detect the [Ne III] emission line in the spectrum, and compare the fluxes of this line of a sample of 15 long GRB host galaxies with survey data, showing the long GRB hosts to be comparable to local metal-poor emission line galaxies in their [Ne III] emission. We fit the supernova bump accompanying this burst, and find that it is similar to other long GRB supernova bumps, but fainter. Conclusions: All properties of GRB 040924 (the associated supernova, the spectrum and SED of host and afterglow) are consistent with an origin in the core-collapse of a massive star.