We perform large-scale density matrix renormalization group simulations of the SU(4) Heisenberg model on the honeycomb lattice to address the long-standing question of its ground state in an unbiased and quantitatively controlled manner. We find reliable numerical evidence that the ground state is a gapped spin-orbital liquid, presumably with a Z4 topological order, characterized by a finite topological entanglement entropy close to ln(4), the absence of both SU(4) and lattice symmetry breaking, and a variationally optimized ground-state energy well below the previously proposed π- flux variational state. By exploiting full SU(4) symmetry and keeping up to 12,800 SU(4) multiplets, corresponding to more than one million U(1) states, we achieve unprecedented accuracy for two- dimensional SU(4) quantum magnets. Finite-size scaling of energies and entanglement entropies supports a robust gapped phase in the two-dimensional limit, while a gapless critical state on narrow cylinders is identified as a proximate remnant of a Dirac spin-orbital liquid. Our results find the SU(4) honeycomb Heisenberg model a realization of a gapped topological spin-orbital liquid and provide convincing numerical evidence for topological order in a highly symmetric two-dimensional quantum magnet.