ReS$_2$ has recently emerged as a new member in the rapidly growing family of two-dimensional materials. Unlike MoS$_2$ or WSe$_2$, the optical and electrical properties of ReS$_2$ are not isotropic due to the reduced symmetry of the crystal. Here, we present layer-dependent Raman measurements of ReS$_2$ samples ranging from monolayers to ten layers in the ultralow frequency regime. We observe layer breathing and shear modes which allow for easy assignment of the number of layers. Polarization-dependent measurements give further insight into the crystal structure and reveal an energetic shift of the shear mode which stems from the in-plane anisotropy of the shear modulus in this material.