The notion of many-body quantum scars is associated with special eigenstates, usually concentrated in certain parts of Hilbert space, that give rise to robust persistent oscillations in a regime that globally exhibits thermalization. Here we extend these studies to many-body systems possessing a true classical limit characterized by a high-dimensional chaotic phase space, which are not subject to any particular dynamical constraint. We demonstrate genuine quantum scarring of wave functions concentrated in the vicinity of unstable classical periodic mean-field modes in the paradigmatic Bose-Hubbard model. These peculiar quantum many-body states exhibit distinct phase-space localization about those classical modes. Their existence is consistent with Heller's scar criterion and appears to persist in the thermodynamic long-lattice limit. Launching quantum wave packets along such scars leads to observable long-lasting oscillations, featuring periods that scale asymptotically with classical Lyapunov exponents, and displaying intrinsic irregularities that reflect the underlying chaotic dynamics, as opposed to regular tunnel oscillations.