Analyzing the spatial organization of molecules in cells and tissues is a cornerstone of biological research and clinical practice. However, despite enormous progress in molecular profiling of cellular constituents, spatially mapping them remains a disjointed and specialized machinery-intensive process, relying on either light microscopy or direct physical registration. Here, we demonstrate DNA microscopy, a distinct imaging modality for scalable, optics-free mapping of relative biomolecule positions. In DNA microscopy of transcripts, transcript molecules are tagged in situ with randomized nucleotides, labeling each molecule uniquely. A second in situ reaction then amplifies the tagged molecules, concatenates the resulting copies, and adds new randomized nucleotides to uniquely label each concatenation event. An algorithm decodes molecular proximities from these concatenated sequences and infers physical images of the original transcripts at cellular resolution with precise sequence information. Because its imaging power derives entirely from diffusive molecular dynamics, DNA microscopy constitutes a chemically encoded microscopy system.