The GW method is routinely used to predict charged valence excitations in molecules and solids. However, the numerical techniques employed in the most efficient GW algorithms break down when computing core excitations as measured by X-ray photoelectron spectroscopy (XPS). We present a full-frequency approach on the real axis using a localized basis to enable the treatment of core levels in GW. Our scheme is based on the contour deformation technique and allows for a precise and efficient calculation of the self-energy, which has a complicated pole structure for core states. The accuracy of our method is validated by comparing to a fully analytic GW algorithm. Furthermore, we report the obtained core-level binding energies and their deviations from experiment for a set of small molecules and large polycyclic hydrocarbons. The core-level excitations computed with our GW approach deviate by less than 0.5 eV from the experimental reference. For comparison, we also report core-level binding energies calculated by density functional theory (DFT)-based approaches such as the popular delta self-consistent field (ΔSCF) method. Our implementation is optimized for massively parallel execution, enabling the computation of systems up to 100 atoms.