A search for Higgs-boson pair production in the bb¯bb¯ final state is carried out with 3.2  fb−1 of proton-proton collision data collected at √s=13  TeV with the ATLAS detector. The data are consistent with the estimated background and are used to set upper limits on the production cross section of Higgs-boson pairs times branching ratio to bb¯bb¯ for both nonresonant and resonant production. In the case of resonant production of Kaluza-Klein gravitons within the Randall-Sundrum model, upper limits in the 24 to 91 fb range are obtained for masses between 600 and 3000 GeV, at the 95% confidence level. The production cross section times branching ratio for nonresonant Higgs-boson pairs is also constrained to be less than 1.22 pb, at the 95% confidence level.