Fabrics—materials consisting of layers of woven fibres—are some of the most important materials in everyday life. Previous nanoscale weavesinclude isotropic crystalline covalent organic frameworks that feature rigid helical strands interlaced in all three dimensions, rather than the two-dimensional layers of flexible woven strands that give conventional textiles their characteristic flexibility, thinness, anisotropic strength and porosity. A supramolecular two-dimensional kagome weave and a single-layer, surface-supported, interwoven two-dimensional polymer have also been reported. The direct, bottom-up assembly of molecular building blocks into linear organic polymer chains woven in two dimensions has been proposed on a number of occasions, but has not previously been achieved. Here we demonstrate that by using an anion and metal ion template, woven molecular ‘tiles’ can be tessellated into a material consisting of alternating aliphatic and aromatic segmented polymer strands, interwoven within discrete layers. Connections between slowly precipitating pre-woven grids, followed by the removal of the ion template, result in a wholly organic molecular material that forms as stacks and clusters of thin sheets—each sheet up to hundreds of micrometres long and wide but only about four nanometres thick—in which warp and weft single-chain polymer strands remain associated through periodic mechanical entanglements within each sheet. Atomic force microscopy and scanning electron microscopy show clusters and, occasionally, isolated individual sheets that, following demetallation, have slid apart from others with which they were stacked during the tessellation and polymerization process. The layered two-dimensional molecularly woven material has long-range order, is birefringent, is twice as stiff as the constituent linear polymer, and delaminates and tears along well-defined lines in the manner of a macroscopic textile. When incorporated into a polymer-supported membrane, it acts as a net, slowing the passage of large ions while letting smaller ions through.