We show that a DNA-based chemical system can be constructed such that it closely approximates the dynamic behavior of an arbitrary system of coupled chemical reactions. Using strand displacement reactions as a primitive we explicitly construct reaction cascades with effectively unimolecular and bimolecular kinetics. Our construction allows for individual reactions to be coupled in arbitrary ways such that reactants can participate in multiple reactions simultaneously, correctly reproducing the desired dynamical properties. Thus arbitrary systems of chemical equations can be compiled into chemistry. We illustrate our method on a chaotic Rössler attractor; simulations of the attractor and of our proposed DNA-based implementation show good agreement.