  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.
