Stable topological defects of light (pseudo)scalar fields can contribute to the Universe's dark energy and dark matter. Currently, the combination of gravitational and cosmological constraints provides the best limits on such a possibility. We take an example of domain walls generated by an axionlike field with a coupling to the spins of standard-model particles and show that, if the galactic environment contains a network of such walls, terrestrial experiments aimed at the detection of wall-crossing events are realistic. In particular, a geographically separated but time-synchronized network of sensitive atomic magnetometers can detect a wall crossing and probe a range of model parameters currently unconstrained by astrophysical observations and gravitational experiments.