The use of square and compass is a practical way of drawing and not a theoretical one, that is, measurements, transports and traces have an action limit that does not exceed one hundredth of a millimeter of accuracy in any case, for each measurement taken.
Therefore and in practice with ruler and compass, in no case will we be able to execute a figure with an error of less than 0.01 mm.
But we can also abstract from practical drawing and immerse ourselves in axiomatic and abstract mathematics and act as hypotheses also with ideal or perfect drawing tools (which do not exist) such as the ideal ruler and compass that can measure with scope and infinite and absolute perfection.
In this hypothetical and axiomatic cases, apart from the ruler and the compass we must in addition to completing them also with a perfect numerical measurement method, which will show us, through appropriate adjustments, if the perfect quadrature can be achieved.
And this does not depend on the measurements of the ruler and the compass, but on the adjustment method giving us the exact result required: No practical measurement with a ruler and compass is necessary, only theoretical measurements and correct mathematical adjustments occur according to the method to follow.
In other words, it is the mathematical adjustment method that we use that will tell us if it can be achieved or not, but it can never be demonstrated in practice with just a ruler and compass.
But of course, we must not forget that we live in a practical and real world, and that drawing tools (ruler, compass, pencil, traces, lines, etc.) are also real and have their limitations.
And any stubborn and obstinate person could argue that if any kind of scale and measurements with the ruler and compass are forbidden to squaring the circle, neither can abstract demonstrations be made with rules and compass that are also ideal and abstract.
That is, the demonstrations or refutations have to be done with the same tools with which the practical executions are carried out, that is, with a ruler and a physical compass.
Therefore and at this point, we could consider two somewhat different ambits:
1.- The field of pure, abstract mathematics and of theoretical methods and adjustments.
2.- The practical field (in our case, the squaring of the circle), of practical squaring with a physical and real ruler and compass.
In the first case of pure and abstract mathematics, an exact fitting method for squaring the circle has not yet been found.
In the second case it can be carried out, but of course exposing at the same time a numerical adjustment that tells us the theoretical accuracy of the method used (due to it can be multiple forms of practical squaring).
And the example here exposed, a squaring is made based on the Phi number, which in my opinion is valid since the inaccuracy that it entails cannot be observed or measured with the work and drawing tools that we have at our disposal.
Thus, by means of the Phi number in this example, we can squaring the circle (theoretically) with an approximation of six decimal digits, (side of the square = 1.77245 | 9139), but as we have said, in the practice of ruler and compass we will only achieve an accuracy of (only 1,772 ... at most) because our vision and drawing tools do not allow us more.
But as the accuracy depends on the fewest number operations with arcs and segment we have to do to get the quadrature, and then, the more operations we do, the less approximation we will have.

I think this example is interesting since we not only squaring the circle with a great theoretical approximation, but we also use the Phi number interrelated with Pi, where we also observe the dependence that these numbers or parameters have between them: Phi and Pi: relation formula

Thanks all you.