Control of magnetism on the atomic scale is becoming essential as data storage devices are miniaturized. We show that antiferromagnetic nanostructures, composed of just a few Fe atoms on a surface, exhibit two magnetic states, the Neel states, that are stable for hours at low temperature. For the smallest structures, we observed transitions between Neel states due to quantum tunneling of magnetization. We sensed the magnetic states of the designed structures using spin-polarized tunneling and switched between them electrically with nanosecond speed. Tailoring the properties of neighboring antiferromagnetic nanostructures enables a low-temperature demonstration of dense nonvolatile storage of information.