(Abridged) Context. NGC 253 is one of only two starburst galaxies found to emit $\gamma$-rays from hundreds of MeV to multi-TeV energies. Accurate measurements of the very-high-energy (VHE) (E $>$ 100 GeV) and high-energy (HE) (E $>$ 60 MeV) spectra are crucial to study the underlying particle accelerators and cosmic-ray interaction and transport.Aims. The measurement of the VHE $\gamma$-ray emission of NGC 253 published in 2012 by H.E.S.S. was limited by large systematic uncertainties. Here, a measurement of the $\gamma$-ray spectrum of NGC 253 is investigated in both HE and VHE $\gamma$-rays.Methods. The data of H.E.S.S. observations are reanalysed using an updated calibration and analysis chain. The $Fermi$-LAT analysis employs more than 8 years of data processed using pass 8. The cosmic-ray particle population is evaluated from the combined HE–VHE $\gamma$-ray spectrum using NAIMA.Results. The VHE $\gamma$-ray energy spectrum is best fit by a power-law with a flux normalisation of $(1.34\,\pm\,0.14^{\mathrm{stat}}\,\pm\,0.27^{\mathrm{sys}}) \times 10^{-13} \mathrm{cm^{-2} s^{-1} TeV^{-1}}$ at 1 TeV — about 40 \% above, but compatible with the value obtained in Abramowski et al. (2012). The spectral index $\Gamma = 2.39 \pm 0.14^{\mathrm{stat}} \pm 0.25^{\mathrm{sys}}$ is slightly softer than but consistent with the previous measurement. At energies above $\sim$3 GeV the HE spectrum is consistent with a power-law ranging into the VHE part of the spectrum measured by H.E.S.S.Conclusions. Two scenarios for the starburst nucleus are tested, in which the gas in the starburst nucleus acts as a target for hadronic cosmic rays. In these two models, the level to which NGC\,253 acts as a calorimeter is estimated to a range of $f_{\rm cal} = 0.1$ to $1$ while accounting for the measurement uncertainties.
