We present early optical photometry and spectroscopy of the afterglow and host galaxy of the bright short-duration gamma-ray burst GRB 130603B discovered by the Swift satellite. Using our Target of Opportunity program on the Gemini South telescope, our prompt optical spectra reveal a strong trace from the afterglow superimposed on continuum and emission lines from the z = 0.3568 +/- 0.0005 host galaxy. The combination of a relatively bright optical afterglow (r' = 21.52 at Delta t = 8.4 hr), together with an observed offset of 0 ''.9 from the host nucleus (4.8 kpc projected distance at z = 0.3568), allow us to extract a relatively clean spectrum dominated by afterglow light. Furthermore, the spatially resolved spectrum allows us to constrain the properties of the explosion site directly, and compare these with the host galaxy nucleus, as well as other short-duration GRB host galaxies. We find that while the host is a relatively luminous (L approximate to 0.8 L-B*), star-forming (SFR = 1.84 M-circle dot yr(-1)) galaxy with almost solar metallicity, the spectrum of the afterglow exhibits weak Ca II absorption features but negligible emission features. The explosion site therefore lacks evidence of recent star formation, consistent with the relatively long delay time distribution expected in a compact binary merger scenario. The star formation rate (SFR; both in an absolute sense and normalized to the luminosity) and metallicity of the host are both consistent with the known sample of short-duration GRB hosts and with recent results which suggest GRB 130603B emission to be the product of the decay of radioactive species produced during the merging process of a neutron-star-neutron-star binary ("kilonova"). Ultimately, the discovery of more events similar to GRB 130603B and their rapid follow-up from 8 m class telescopes will open new opportunities for our understanding of the final stages of compact-objects binary systems and provide crucial information (redshift, metallicity, and chemical content of their explosion site) to characterize the environment of one of the most promising gravitational wave sources.