Traditionally, assembly lines to build machines, from electronic circuits to motor vehicles, follow specific instruction manuals, followed by robots or people. On the other hand, in biology, organisms self-assemble spontaneously according to instructions encoded in their genes - nonetheless following the laws of physics. Inspired by biology, we design and develop emulsion droplets with specific DNA interactions that guide their spontaneous assembly into linear or branched freely-jointed polymers. In addition, we introduce and trigger secondary DNA interactions along the polymer backbone to fold it into several simple yet distinguishable structures. This physical system demonstrates the possibility of controlling the self-assembly of monomer droplets into polymers, which in turn allows us to study their collapse and folding to explore the underlying free energy landscape. Unlike molecular polymers, colloidomers can be assembled and disassembled at will, giving rise to reconfigurable materials. The droplets can readily be solidified; therefore they offer a route to hands-off manufacturing of objects with inbuilt hierarchies.