MAXI J1305-704 has been proposed as a high-inclination candidate black hole X-ray binary in view of its X-ray properties and dipping behaviour during outburst. We present photometric and spectroscopic observations of the source in quiescence that allow us to reveal the ellipsoidal modulation of the companion star and absorption features consistent with those of an early K-type star ($T_{\rm eff}=4610^{+130}_{-160}\, {\rm K}$). The central wavelengths of the absorption lines vary periodically at $P_{\rm orb}=0.394\pm 0.004\, {\rm d}$ with an amplitude of $K_2=554\pm 8\, {\rm km \, s^{-1}}$. They imply a mass function for the compact object of $f(M_1)=6.9\pm 0.3\, {\rm M}_\odot$, confirming its black hole nature. The simultaneous absence of X-ray eclipses and the presence of dips set a conservative range of allowed inclinations $60\, {\rm deg}\lt i\lt 82\, {\rm deg}$, while modelling of optical light curves further constrain it to $i=72^{+5}_{-8}\, {\rm deg}$. The above parameters together set a black hole mass of $M_1= 8.9_{-1.0}^{+1.6}\, {\rm M}_{\odot }$ and a companion mass of $M_2= 0.43\pm 0.16\, {\rm M}_{\odot }$, much lower than that of a dwarf star of the observed spectral type, implying it is evolved. Estimates of the distance to the system ($d=7.5^{+1.8}_{-1.4}\, {\rm kpc}$) and space velocity ($v_{\rm space}=270\pm 60 \, {\rm km\, s^{-1}}$) place it in the Galactic thick disc and favour a significant natal kick during the formation of the BH if the supernova occurred in the Galactic Plane.