Measurements of differential and integrated fiducial cross-sections in events with two same-flavor, opposite-sign electron or muon pairs are presented with the data corresponding to 139 fb$^{-1}$ of $\sqrt{s} = 13$ TeV proton--proton collisions collected by the ATLAS dectector during Run 2 of LHC operations (2015--2018). The differential cross-sections exhibit a rich structure with contributions from a number of Standard Model processes. The measurements are performed and compared to state-of-the-art Standard Model calculations, which are found to be consistent with data. The branching ratio of the rare $Z\rightarrow4\ell$ decay is extracted. Constraints on a new physics model based on a spontaneously broken B-L gauge symmetry are also evaluated.