CUPID-0 is the first large mass array of enriched Zn$^{82}$Se scintillating low temperature calorimeters, operated at LNGS since 2017. During its first scientific runs, CUPID-0 collected an exposure of 9.95 kg yr. Thanks to the excellent rejection of $alpha$ particles, we attained the lowest background ever measured with thermal detectors in the energy region where we search for the signature of $^{82}$Se neutrinoless double beta decay. In this work we develop a model to reconstruct the CUPID-0 background over the whole energy range of experimental data. We identify the background sources exploiting their distinctive signatures and we assess their extremely low contribution (down to $sim10^{-4}$ counts/(keV kg yr)) in the region of interest for $^{82}$Se neutrinoless double beta decay search. This result represents a crucial step towards the comprehension of the background in experiments based on scintillating calorimeters and in next generation projects such as CUPID.
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