Combining ferroelectricity with other properties such as visible light absorption or long-range magnetic order requires the discovery of new families of ferroelectric materials. Here, through the analysis of a high-throughput database of phonon band structures, we identify a structural family of anti–Ruddlesden–Popper phases A4X2O (A=Ca, Sr, Ba, Eu, X=Sb, P, As, Bi) showing ferroelectric and antiferroelectric behaviors. The discovered ferroelectrics belong to the new class of hyperferroelectrics that polarize even under open-circuit boundary conditions. The polar distortion involves the movement of O anions against apical A cations and is driven by geometric effects resulting from internal chemical strains. Within this structural family, we show that Eu4Sb2O combines coupled ferromagnetic and ferroelectric order at the same atomic site, a very rare occurrence in materials physics.