  A search for nonresonant Higgs boson pair production in the
$b\bar{b}b\bar{b}$ final state is presented. The analysis uses 126 fb$^{-1}$ of
$pp$ collision data at $\sqrt{s}={13}$ 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.
