Modern materials can be designed to self-organize into structures that perform a wide range of functions. The past two decades have witnessed major innovations in the versatility of building blocks, ranging from DNA on the nanoscale to handshaking materials on the macroscale. One can reliably self-assemble arbitrary structures like a jigsaw puzzle if all the pieces are distinct. Still, systems with fewer flavors of building blocks have so far been limited to the assembly of exotic crystals. Inspired by Nature’s strategy of folding biopolymers into specific RNA and protein structures, here we introduce a minimal model system of colloidal polymers with programmable DNA interactions that guide their downhill folding into two-dimensional geometries. Combining experiments, simulations, and theory, we show that designing the order in which interactions are switched on directs folding into unique geometries called foldamers. These foldamers can further interact to make complex supracolloidal architectures, seeding the next generation of bio-inspired materials. Our results are independent of the dynamics and therefore apply to polymeric materials with hierarchical interactions on all length scales, from organic molecules all the way to Rubik’s snakes.