The burden is on the specifier of the compass and straight-edge algorithm to explain a reasonable mechanism for carrying out the steps exactly, to infinite precision. Without this, the procedure can only be expected to give a reliable approximation of sqrt(2) to some finite precision, which a Turing machine can match. A similar observation applies to chaotic systems.
In general, a chaotic system can only be said to "solve" a problem if it can be configured to reliably arrive at (an encoding of) the solution for a given instance, despite any chaotic behaviour exhibited by the system. It wouldn't be reasonable to, for example, claim that a Turing machine cannot solve the problem of predicting the behaviour of a chaotic system with some particular initial conditions, if you couldn't reasonably conifgure the chaotic system itself to do that either.