We have used angle resolved photoemission spectroscopy to investigate the band structure of ReS$_2$, a transition-metal dichalcogenide semiconductor with a distorted 1T crystal structure. We find a large number of narrow valence bands, which we attribute to the combined influence of the structural distortion and spin-orbit coupling. We further image how this leads to a strong in-plane anisotropy of the electronic structure, with quasi-one-dimensional bands reflecting predominant hopping along zig-zag Re chains. We find that this does not persist up to the top of the valence band, where a more three-dimensional character is recovered with the fundamental band gap located away from the Brillouin zone centre along $k_z$. These experiments are in good agreement with our density-functional theory calculations, shedding new light on the bulk electronic structure of ReS$_2$, and how it can be expected to evolve when thinned to a single layer.