Obviously, with a physical straightedge, compass, and pencil, no construction can be carried out to infinite accuracy; what is required here is a construction that would be to infinite accuracy, assuming that all the tools work to infinite accuracy.
In other words, it is not good enough to get an answer that is accurate to one part in a million or to one part in a trillion--an infinitely accurate result is required. As we will see later, if you ``only'' need accuracy to one part in a trillion, you can construct anything with a straightedge and compass.
Finally, the construction has to be completed in a finite number of steps; you cannot require an infinite set of operations to construct a point that is the limit of these operations. If an infinite number of steps were allowed, again, anything can be constructed. Simply construct points to an accuracy of one in a million, then one in a billion, then one in a trillion, and continue forever. The limit of these points will be the exact solution, but only after an infinite number of steps.