Topological states of matter lie at the heart of our modern understanding of condensed matter systems. In two-dimensional (2D) quantum Hall insulators, the non-trivial topology, defined by the first Chern number, manifests as a quantized Hall conductance and protected ballistic edge modes. Besides topological insulators and Weyl semi-metals experimentally realized in 3D materials, a large variety of topological systems, theoretically predicted in dimensions $D > 3$, remains unexplored - among them a generalization of the quantum Hall effect in 4D. So far, topological properties linked with the 4D Hall effect have been revealed via geometrical charge pump experiments in 2D systems. A truly 4D Hall system has also been realized using electronic circuits - however, no direct evidence of topological quantization has been reported. Here, we engineer an atomic quantum Hall system evolving in 4D, by coupling with light fields two spatial dimensions and two synthetic ones encoded in the electronic spin $J = 8$ of dysprosium atoms. We measure the characteristic properties of a 4D quantum Hall system, namely the quantization of its electromagnetic non-linear response by a second Chern number, and the special nature of its 3D hyperedge modes, which combine ballistic motion along one orientation and insulating behaviour in the two remaining directions. We also probe low-lying excitations, revealing non-planar cyclotron orbits in contrast with their circular equivalents in $D\leq3$. Our findings pave the way to the exploration of interacting quantum Hall systems in 4D, from the investigation of strongly-correlated liquids to the simulation of high-energy models in link with quantum gravity and Yang-Mills field theory.