New information on short/hard gamma-ray bursts (GRBs) is being gathered thanks to the discovery of their optical and X-ray afterglows. However, some key aspects are still poorly understood, including the collimation level of the outflow, the duration of the central engine activity, and the properties of the progenitor systems. We want to constrain the physical properties of the short GRB 050724 and of its host galaxy, and make some inferences on the global short GRB population. We present optical observations of the afterglow of GRB 050724 and of its host galaxy, significantly expanding the existing dataset for this event. We compare our results with models, complementing them with available measurements from the literature. We study the afterglow light curve and spectrum including X-ray data. We also present observations of the host galaxy. The observed optical emission was likely related to the large flare observed in the X-ray light curve. The apparent steep decay was therefore not due to the jet effect. Available data are indeed consistent with low collimation, in turn implying a large energy release, comparable to that of long GRBs. The flare properties also constrain the internal shock mechanism, requiring a large Lorentz factor contrast between the colliding shells. This implies that the central engine was active at late times, rather than ejecting all shells simultaneously. The host galaxy has red colors and no ongoing star formation, consistent with previous findings on this GRB. However, it is not a pure elliptical, and has some faint spiral structure. GRB 050724 provides the most compelling case for association between a short burst and a galaxy with old stellar population. It thus plays a pivotal role in constraining progenitors models, which should allow for long delays between birth and explosion.
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