The inclusive top-quark pair (tt¯) production cross-section σ_tt¯ is measured in proton–proton collisions at a centre-of-mass energy s = 5.02 TeV, using 257 pb⁻¹ of data collected in 2017 by the ATLAS experiment at the LHC. The tt¯ cross-section is measured in both the dilepton and single-lepton final states of the tt¯ system and then combined. The combination of the two measurements yields σtt¯=67.5±0.9(stat.)±2.3(syst.)±1.1(lumi.)±0.2(beam)pb, where the four uncertainties reflect the limited size of the data sample, experimental and theoretical systematic effects, and imperfect knowledge of both the integrated luminosity and the LHC beam energy, giving a total uncertainty of 3.9%. The result is in agreement with theoretical quantum chromodynamic calculations at next-to-next-to-leading order in the strong coupling constant, including the resummation of next-to-next-to-leading logarithmic soft-gluon terms, and constrains the parton distribution functions of the proton at large Bjorken-x. [Figure not available: see fulltext.]