Natural biomolecular assemblies such as molecular motors, enzymes, viruses and subcellular structures often form by self-limiting hierarchical oligomerization of multiple subunits. Large structures can also assemble efficiently from a few components by combining hierarchical assembly and symmetry, a strategy exemplified by viral capsids. De novo protein design and RNA and DNA nanotechnology aim to mimic these capabilities, but the bottom-up construction of artificial structures with the dimensions and complexity of viruses and other subcellular components remains challenging. Here we show that natural assembly principles can be combined with the methods of DNA origami to produce gigadalton-scale structures with controlled sizes. DNA sequence information is used to encode the shapes of individual DNA origami building blocks, and the geometry and details of the interactions between these building blocks then control their copy numbers, positions and orientations within higher-order assemblies. We illustrate this strategy by creating planar rings of up to 350 nanometres in diameter and with atomic masses of up to 330 megadaltons, micrometre-long, thick tubes commensurate in size to some bacilli, and three-dimensional polyhedral assemblies with sizes of up to 1.2 gigadaltons and 450 nanometres in diameter. We achieve efficient assembly, with yields of up to 90 per cent, by using building blocks with validated structure and sufficient rigidity, and an accurate design with interaction motifs that ensure that hierarchical assembly is self-limiting and able to proceed in equilibrium to allow for error correction. We expect that our method, which enables the self-assembly of structures with sizes approaching that of viruses and cellular organelles, can readily be used to create a range of other complex structures with well defined sizes, by exploiting the modularity and high degree of addressability of the DNA origami building blocks used.