The association of some short-duration gamma-ray bursts (GRBs) with elliptical galaxies established that their progenitors, unlike those of long GRBs, belong to an old stellar population. However, the majority of short GRBs appear to occur in star forming galaxies, raising the possibility that some progenitors are related to recent star formation activity. Here, we present optical spectroscopy of these hosts and measure their luminosities, star formation rates, and metallicities. We find luminosities of L-B approximate to 0.1-1.5 L-*, star formation rates of SFR approximate to 0.2-6 M-circle dot yr(-1), and metallicities of 12 + log(O/H) approximate to 8.5-8.9 (Z approximate to 0.6-1.6 Z(circle dot)). A detailed comparison to the hosts of long GRBs reveals systematically higher luminosities, lower specific star formation rates (SFR/LB) by about an order of magnitude, and higher metallicities by about 0.6 dex. The Kolmogorov-Smirnov probability that the short and long GRB hosts are drawn from the same underlying galaxy distribution is only similar to 10(-3). Short GRB hosts exhibit excellent agreement with the specific star formation rates and the luminosity-metallicity relation of field galaxies at z similar to 0.1-1. We thus conclude that short GRB hosts are not dominated by young stellar populations like long GRB hosts. Instead, short GRB hosts appear to be drawn uniformly from the underlying field galaxy distribution, indicating that the progenitors have a wide age distribution of several Gyr.