SABRE (Sodium-iodide with Active Background REjection) is a direct dark matter search experiment based on an array of radio-pure NaI(Tl) crystals surrounded by a liquid scintillator veto. Twin SABRE experiments in the Northern and Southern Hemispheres will differentiate a dark matter signal from seasonal and local effects. The experiment is currently in a Proof-of-Principle (PoP) phase, whose goal is to demonstrate that the background rate is low enough to carry out an independent search for a dark matter signal, with sufficient sensitivity to confirm or refute the DAMA result during the following full-scale experimental phase. The impact of background radiation from the detector materials and the experimental site needs to be carefully investigated, including both intrinsic and cosmogenically activated radioactivity. Based on the best knowledge of the most relevant sources of background, we have performed a detailed Monte Carlo study evaluating the expected background in the dark matter search spectral region. The simulation model described in this paper guides the design of the full-scale experiment and will be fundamental for the interpretation of the measured background and hence for the extraction of a possible dark matter signal.