A measurement of four-top-quark production using proton-proton collision data at a centre-of-mass energy of 13 TeV collected by the ATLAS detector at the Large Hadron Collider corresponding to an integrated luminosity of 139 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ is presented. Events are selected if they contain a single lepton (electron or muon) or an opposite-sign lepton pair, in association with multiple jets. The events are categorised according to the number of jets and how likely these are to contain $b$-hadrons. A multivariate technique is then used to discriminate between signal and background events. The measured four-top-quark production cross section is found to be 26$\mathrm{<mrow\; data-mjx-texclass="ORD">-15</mrow><mrow\; data-mjx-texclass="ORD">+17</mrow>}$ fb, with a corresponding observed (expected) significance of 1.9 (1.0) standard deviations over the background-only hypothesis. The result is combined with the previous measurement performed by the ATLAS Collaboration in the multilepton final state. The combined four-top-quark production cross section is measured to be 24$\mathrm{<mrow\; data-mjx-texclass="ORD">-6</mrow><mrow\; data-mjx-texclass="ORD">+7</mrow>}$ fb, with a corresponding observed (expected) signal significance of 4.7 (2.6) standard deviations over the background-only predictions. It is consistent within 2.0 standard deviations with the Standard Model expectation of 12.0$\pm$2.4 fb.