In strongly correlated electron systems periodic modulations on the nano-scale have typically been associated with competition between short- and long-range interactions, for example, between exchange and dipole-dipole interactions in the case of ferromagnetic thin films. Here we show that spin-stripe textures may develop also in antiferromagnets, where long-range dipole-dipole magnetic interactions are absent. We discover a magnetic stripe structure which appears at the transition between the spiral and collinear magnetic orders in β-TeVO4 compound, a nearly perfect realization of a frustrated (zigzag) ferromagnetic spin-1/2 chain. The presented system thus allows better understanding of the origin of the intriguing nanometer-sized modulation in other systems, e.g., high-temperature superconductors.