We study the large-mass limit of interacting quantum (Bose or Fermi) gases in thermal equilibrium. We show that in the suitably-defined large-mass limit, the system gives rise to a gas of classical interacting particles. The corresponding question for bosons on a lattice was previously addressed by Fröhlich, Knowles, Schlein, and the third author. In this work, we study the continuum regime which requires us to suitably tune the chemical potential. The starting point of our analysis is the Ginibre loop ensemble, which allows one to describe a system of interacting quantum gases in thermal equilibrium in terms of an ensemble of interacting Brownian paths. In a finite volume, our analysis is performed for stable and Hölder continuous interaction potentials and we are able to obtain explicit rates of convergence. When the interaction potential is nonnegative and satisfies suitable integrability conditions, we study the associated infinite-volume problem by means of cluster expansions.