Large scale modeling of condensed phases of elemental materials using atomistic molecular dynamics is necessary for studying phase transformations, dynamical, and material properties. The challenge lies in developing realistic inter-atomic potentials. Embedded-atom type many-body models are a promising class of potentials to be explored. In particular, a many-body model for describing rare-gas solids is proposed. The model is used to calculate the pressure-dependent elastic constants, Cauchy discrepancies, and melting curves of Ne, Ar, Kr, and Xe. It is shown that a spherically symmetric many-body model cannot reproduce the high-pressure elastic properties of the system. The model successfully reproduces the experimentally observed negative Cauchy discrepancies and improves the agreement of the melting curves of rare-gas solids compared to pair-interacting calculations.