A search for resonant Higgs boson pair production in the $bb¯bb¯$ final state is presented. The analysis uses 126-139 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ 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.