Strontium optical lattice clocks have the potential to simultaneously interrogate millions of atoms with a spectroscopic quality factor Q~4e17. Previously, atomic interactions have forced a compromise between clock stability, which benefits from a large atom number, and accuracy, which suffers from density-dependent frequency shifts. Here, we demonstrate a scalable solution which takes advantage of the high, correlated density of a degenerate Fermi gas in a three-dimensional optical lattice to guard against on-site interaction shifts. Using a state-of-the-art ultra-stable laser, we achieve an unprecedented level of atom-light coherence, reaching Q = 5.2e15 with 1e4 atoms. We investigate clock systematics unique to this design; in particular, we show that contact interactions are resolved so that their contribution to clock shifts is orders of magnitude lower than in previous experiments, and we measure the combined scalar and tensor magic wavelengths for state-independent trapping along all three lattice axes.