The host of the short gamma-ray burst (GRB) 050709 is a morphologically disturbed low-luminous galaxy. At a redshift of z = 0.16, it belongs to one of the cosmologically nearest short-GRB hosts identified to date. Consequently, it represents a promising target for sensitive, spatially resolved observational studies. We have used the Multi Unit Spectroscopic Explorer (MUSE) mounted at the Very Large Telescope to characterize the GRB host galaxy. In addition, we performed deep radio-continuum observations of the host with the Australia Telescope Compact Array (ATCA) and with ALMA at 1.3 mm. Moreover, we made use of archival Spitzer Space Telescope 24 μm and Hubble Space Telescope/F814W imaging data of this galaxy. The spatially resolved MUSE data reveal that the entire host is a source of strong line emission, in particular from Hα and [O III] λ 5007, superimposed on a rather weak stellar continuum. Using the Balmer decrement, we map the internal host-galaxy reddening and derive an extinction-corrected star formation rate based on the flux in the Hα line of 0.15 ± 0.02 M⊙ yr−1. The galaxy is detected neither by ALMA nor by Spitzer, excluding a substantial amount of optically obscured star formation activity. Using the O3N2 metallicity indicator, we measure an average 12 + log (O/H) = 8.40 ± 0.05 (corresponding to ∼0.5 solar). Diagnostic emission line diagrams show that a substantial fraction of all MUSE spaxels that cover the GRB 050709 host galaxy lie close to the star-formation demarcation line. Some spaxels even suggest line emission by shocked gas. The ATCA observations reveal faint diffuse radio emission at the eastern part of the host in excess to that expected from pure star formation, possibly further evidence for nonthermal processes. The kinematics of the Hα-emitting gas suggests a rotationally supported host-galaxy system, apparently in contrast to its irregular photometric morphology. A comparison with the field-galaxy population reveals, however, that the kinematics of the gas in the 050709 host fits into the ensemble of merging galaxies well. Finally, we use the ATCA radio data to set deep constraints on any late-time flux from the GRB afterglow or a potentially associated kilonova radio flare ∼10 years after the burst.