It has been a long-standing goal to create magnetism in a non-magnetic material by manipulating its structure at the nanoscale. Many structural defects have unpaired spins; an ordered arrangement of these can create a magnetically ordered state. In this article we predict theoretically that stepped silicon surfaces stabilized by adsorbed gold achieve this state by self assembly, creating chains of polarized electron spins with atomically precise structural order. The spins are localized at silicon step edges having the form of graphitic ribbons. The predicted magnetic state is supported by recent experimental observations, such as the coexistence of double- and triple-period distortions and the absence of edge states in photoemission. Ordered arrays of surface spins can be accessed by probes with single-spin sensitivity, such as spin-polarized scanning tunnelling microscopy. The integration of structural and magnetic order is crucial for technologies involving spin-based computation and storage at the atomic level.