Measurements of four-lepton differential and integrated fiducial cross-sections in events with two same-flavour, opposite-charge electron or muon pairs are presented. The data correspond to 139 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ of $s=13$ TeV proton-proton collisions, collected by the ATLAS detector during Run 2 of the Large Hadron Collider (2015-2018). The final state has contributions from a number of interesting Standard Model processes that dominate in different four-lepton invariant mass regions, including single $Z$ boson production, Higgs boson production and on-shell $ZZ$ production, with a complex mix of interference terms, and possible contributions from physics beyond the Standard Model. The differential cross-sections include the four-lepton invariant mass inclusively, in slices of other kinematic variables, and in different lepton flavour categories. Also measured are dilepton invariant masses, transverse momenta, and angular correlation variables, in four regions of four-lepton invariant mass, each dominated by different processes. The measurements are corrected for detector effects and are compared with state-of-the-art Standard Model calculations, which are found to be consistent with the data. The $Z\to 4\ell$ branching fraction is extracted, giving a value of $(4.41\pm 0.30)\times 10-6$. Constraints on effective field theory parameters and a model based on a spontaneously broken $B-L$ gauge symmetry are also evaluated. Further reinterpretations can be performed with the provided information.