Point contacts provide simple connections between macroscopic particle reservoirs. In electric circuits, strong links between metals, semiconductors, or superconductors have applications for fundamental condensed-matter physics as well as quantum information processing. However, for complex, strongly correlated materials, links have been largely restricted to weak tunnel junctions. We studied resonantly interacting Fermi gases connected by a tunable, ballistic quantum point contact, finding a nonlinear current-bias relation. At low temperature, our observations agree quantitatively with a theoretical model in which the current originates from multiple Andreev reflections. In a wide contact geometry, the competition between superfluidity and thermally activated transport leads to a conductance minimum. Our system offers a controllable platform for the study of mesoscopic devices based on strongly interacting matter.