Decays of B0 and B0s mesons to the charmless baryonic final states p¯ph+h'−, where h and h′ each denote a kaon or a pion, are searched for using the LHCb detector. The analysis is based on a sample of proton-proton collision data collected at center-of-mass energies of 7 and 8 TeV, corresponding to an integrated luminosity of 3  fb−1. Four-body charmless baryonic B0s decays are observed for the first time. The decays B0s→p¯pK+K−, B0s→p¯pK±π∓, B0→p¯pK±π∓ and B0→p¯pπ+π− are observed with a significance greater than 5 standard deviations; evidence at 4.1 standard deviations is found for the B0→p¯pK+K− decay and an upper limit is set on the branching fraction for B0s→p¯pπ+π−. Branching fractions in the kinematic region m(p¯p)<2850  MeV/c2 are measured relative to the B0→J/ψ(→p¯p)K∗(892)0 channel.