Compact fusion reactors are receiving increasing interest as a promising route for accelerating the path toward commercial fusion, thanks to their reduced size and cost. However, this compactness introduces new technological challenges, including higher radiation loads on critical functional components, such as the magnet system. Neutron shielding is therefore of utmost importance to guarantee the expected lifetime of the device, and its selection must account for the harsh environment imposed by the high radiation flux. Shielding materials should be structurally stable, not melt within the operational temperature windows, and be relatively low-cost. For nuclear reactor applications, binary compounds are typically the preferred choice as they often meet these requirements, particularly in terms of availability and cost. In this work, we present a systematic Monte Carlo analysis of more than 700 binary compounds, exposed to the neutron spectrum at the most loaded position of the vacuum vessel in a simplified model of a compact fusion reactor. Shielding performances were evaluated in a toroidal geometry in terms of neutron attenuation, power deposition, and activation, leading to the identification of several promising compositions for effective neutron shielding in future fusion applications.