A search for Higgs-boson pair production in the $b\bar{b}b\bar{b}$ final state is carried out with 3.2~$fb^{-1}$ of proton-proton collision data collected at $\sqrt{s} = 13$ 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 times branching ratio to $b\bar{b}b\bar{b}$ for Kaluza-Klein gravitons in the context of the Randall--Sundrum model with a warped extra dimension and for a model with a narrow spin-0 resonance. In the context of the Randall--Sundrum model, upper limits of $\sim70$ fb are obtained over a broad mass range of 600 and 3000 GeV, at the 95% confidence level. The production cross section times branching ratio for non-resonant Higgs-boson pairs is also constrained to be less than 1.22 pb.

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