Antiphase domain boundaries (APDBs) in the $(\sqrt{2}\ifmmode\times\else\texttimes\fi{}\sqrt{2})R{45}^{\ensuremath{\circ}}$ reconstruction of the Fe${}_{3}$O${}_{4}$(001) surface were investigated using scanning tunneling microscopy (STM) and density functional theory [(DFT) + $U$] calculations. The equilibrium structure of the APDBs is interpreted in terms of the distorted $B$-layer model for the $(\sqrt{2}\ifmmode\times\else\texttimes\fi{}\sqrt{2})R{45}^{\ensuremath{\circ}}$ reconstruction in which a lattice distortion couples to charge order in the subsurface layers. The APDBs are observed after prolonged annealing at 700 \ifmmode^\circ\else\textdegree\fi{}C, indicating that they are extremely stable. DFT + $U$ calculations reveal that the APDB structure is linked to a disruption in the subsurface charge-order pattern, leading to an enrichment of Fe${}^{2+}$ cations at the APDB. Simulated STM images reproduce the appearance of the APDBs in the experimental data and reveal that they are preferential adsorption sites for hydrogen atoms.