An autocatalytic binary polymer system is studied as an abstract model for a chemical reaction network capable to evolve. Due to autocatalysis, long polymers appear spontaneously and their concentration is shown to be maintained at the same level as that of monomers. When the reaction starts from a pool of monomers, highly ordered populations with particular sequence patterns are dynamically selected out of a vast number of possible states. The interplay between the selected microscopic sequence patterns and the macroscopic cooperative structures is examined both analytically and in simulation. Stability, fluctuations, and dynamic selection mechanisms are investigated for the involved self-organizing processes.