We present results of an extensive observing campaign of the short- duration, hard spectrum gamma-ray burst (GRB) 070724A, aimed at detecting the radioactively powered emission that might follow from a binary merger or collapse involving compact objects. Our multiband observations span the range in time over which this so-called Li-Paczyŉski mini-supernova (mini-SN) could be active, beginning within 3 h of the GRB trigger and represent some of the deepest and most comprehensive searches for such emission. We find no evidence for such activity and place limits on the abundances and the lifetimes of the possible radioactive nuclides that could form in the rapid decompression of nuclear density matter. Furthermore, our limits are significantly fainter than the peak magnitude of any previously detected broad-lined Type Ic SN associated with other GRBs, effectively ruling out a long GRB-like SN for this event. Given the unambiguous redshift of the host galaxy (z = 0.456), GRB 070724A represents one of a small, but growing, number of short-hard GRBs for which firm physical/rest-frame quantities currently exist. The host of GRB 070724A is a moderately star-forming galaxy with an older stellar population component and a relatively high metallicity of 12 + log(O/H)$_KD02$ = 9.1. We find no significant evidence for large amounts of extinction along the line of sight that could mask the presence of an SN explosion and estimate a small probability for chance alignment with the putative host. We discuss how our derived constraints fit into the evolving picture of short-hard GRBs, their potential progenitors and the host environments in which they are thought to be produced.