A search for resonant Higgs boson pair production in the bb¯bb¯ final state is presented. The analysis uses 126 fb-1-139 fb-1 of pp collision data at s=13 TeV collected with the ATLAS detector at the Large Hadron Collider. The analysis is divided into two channels, targeting Higgs boson decays which are reconstructed as pairs of small-radius jets or as individual large-radius jets. Spin-0 and spin-2 benchmark signal models are considered, both of which correspond to resonant HH production via gluon-gluon fusion. The data are consistent with Standard Model predictions. Upper limits are set on the production cross section times branching ratio to Higgs boson pairs of a new resonance in the mass range from 251 GeV to 5 TeV.