It is not known how life arose from prebiotic physical chemistry. How did fruitful cell-like associations emerge from the two polymer types—informational (nucleic acids, xNAs = DNA or RNA) and functional (proteins)? Our model shows how functional networks could bootstrap from random sequence-independent initial states. For proteins, we adopt the foldamer hypothesis: through persistent nonequilibrium prebiotic syntheses, short random peptides fold and catalyze the elongation of others. The xNAs enter through random binding to the peptides, and all chains can mutate. Chains grow inside colloids that split when they’re large, coupling faster growth speeds to bigger populations. Random and useless at first, these folding and binding events grow protein—xNA networks that resemble today’s protein–protein networks.