A search for the production of three massive vector bosons in proton-proton collisions is performed using data at $\sqrt{s} = 13$ TeV recorded with the ATLAS detector at the Large Hadron Collider in the years 2015-2017, corresponding to an integrated luminosity of $79.8$ fb$^{-1}$. Events with two same-sign leptons $\ell$ (electrons or muons) and at least two reconstructed jets are selected to search for $WWW \to \ell \nu \ell \nu qq$. Events with three leptons without any same-flavour opposite-sign lepton pairs are used to search for $WWW \to \ell \nu \ell\nu \ell \nu$, while events with three leptons and at least one same-flavour opposite-sign lepton pair and one or more reconstructed jets are used to search for $WWZ \to \ell \nu qq \ell \ell$. Finally, events with four leptons are analysed to search for $WWZ \to \ell \nu \ell \nu \ell \ell$ and $WZZ \to qq \ell \ell \ell \ell$. Evidence for the joint production of three massive vector bosons is observed with a significance of 4.1 standard deviations, where the expectation is 3.1 standard deviations.