We present the results of an exploratory lattice QCD calculation of three-baryon systems through a high statistics study of one ensemble of anisotropic clover gauge-field configurations with a pion mass of m π ∼ 390     MeV . Because of the computational cost of the necessary contractions, we focus on correlation functions generated by interpolating operators with the quantum numbers of the Ξ 0 Ξ 0 n system, one of the least demanding three-baryon systems in terms of the number of contractions. We find that the ground state of this system has an energy of E Ξ 0 Ξ 0 n = 3877.9 ± 6.9 ± 9.2 ± 3.3     MeV corresponding to an energy shift due to interactions of δ E Ξ 0 Ξ 0 n = E Ξ 0 Ξ 0 n − 2 M Ξ 0 − M n = 4.6 ± 5.0 ± 7.9 ± 4.2     MeV . There are a significant number of time slices in the three-baryon correlation function for which the signal-to-noise ratio is only slowly degrading with time. This is in contrast to the exponential degradation of the signal-to-noise ratio that is observed at larger times, and is due to the suppressed overlap of the source and sink interpolating operators that are associated with the variance of the three-baryon correlation function onto the lightest eigenstates in the lattice volume (mesonic systems). As one of the motivations for this area of exploration is the calculation of the structure and reactions of light nuclei, we also present initial results for a system with the quantum numbers of the triton ( p n n ). This present work establishes a path to multibaryon systems, and shows that lattice QCD calculations of the properties and interactions of systems containing four and five baryons are now within sight.