Using a combined experimental and theoretical approach, we address C $1s$ core-level shifts in amorphous carbon. Experimental results are obtained by x-ray photoelectron spectroscopy (XPS) and electron-energy-loss spectroscopy (EELS) on thin-film samples of different atomic density, obtained by a pulsed-laser deposition growth process. The XPS spectra are deconvoluted into two contributions, which are attributed to ${\mathrm{sp}}^2$- and ${\mathrm{sp}}^3$-hybridized atoms, respectively, separated by 0.9 eV, independent of atomic density. The ${\mathrm{sp}}^3$ hybridization content extracted from XPS is consistent with the atomic density derived from the plasmon energy in the EELS spectrum. In our theoretical study, we generate several periodic model structures of amorphous carbon of different densities applying two schemes of increasing accuracy in sequence. We first use a molecular-dynamics approach, based on an environmental-dependent tight-binding Hamiltonian to quench the systems from the liquid phase. The final model structures are then obtained by further atomic relaxation using a first-principles pseudopotential plane-wave approach within density-functional theory. Within the latter framework, we also calculate carbon $1s$ core-level shifts for our disordered model structures. We find that the shifts associated to threefold- and fourfold- coordinated carbon atoms give rise to two distinct peaks separated by about 1.0 eV, independent of density, in close agreement with experimental observations. This provides strong support for decomposing the XPS spectra into two peaks resulting from ${\mathrm{sp}}^2$- and ${\mathrm{sp}}^3$-hybridized atoms. Core-hole relaxations effects account for about 30% of the calculated shifts.