A search for nonresonant Higgs boson pair production in the $b\overline{b}b\overline{b}$ final state is presented. The analysis uses $126{\mathrm{fb}}^{-1}$ of $pp$ collision data at $\sqrt{s}=13\mathrm{TeV}$ collected with the ATLAS detector at the Large Hadron Collider, and targets both the gluon-gluon fusion and vector-boson fusion production modes. No evidence of the signal is found and the observed (expected) upper limit on the cross section for nonresonant Higgs boson pair production is determined to be 5.4 (8.1) times the Standard Model predicted cross section at 95% confidence level. Constraints are placed on modifiers to the $HHH$ and $HHVV$ couplings. The observed (expected) $2\sigma$ constraints on the $HHH$ coupling modifier, ${\kappa }_{\lambda }$, are determined to be $[-3.5,11.3]$ ($[-5.4,11.4]$), while the corresponding constraints for the $HHVV$ coupling modifier, ${\kappa }_{2V}$, are $[-0.0,2.1]$ ($[-0.1,2.1]$). In addition, constraints on relevant coefficients are derived in the context of the Standard Model effective field theory and Higgs effective field theory, and upper limits on the $HH$ production cross section are placed in seven Higgs effective field theory benchmark scenarios.