The problem is old and precisely stated: given a circle, construct with compass and straightedge a square of exactly equal area. Plutarch reports that Anaxagoras worked on it in prison in the fifth century BC, which makes it one of the oldest recorded open problems in mathematics. It stayed open for more than two thousand years.
It is now closed, and closed by proof rather than by exhaustion. The chain runs as follows. Squaring the circle requires constructing a length of sqrt pi from a unit length, which in turn requires constructing pi. In 1761 Johann Heinrich Lambert proved that pi is irrational: it is not the ratio of two whole numbers. That alone does not settle the question, because many irrational lengths are perfectly constructible; sqrt 2 is simply the diagonal of the unit square. The decisive result came in 1882, when Ferdinand von Lindemann proved pi transcendental, meaning it is not a root of any polynomial equation with rational coefficients.
That finishes it. Every length reachable by compass and straightedge is algebraic, and its degree over the rationals is a power of 2, because each new point arises from intersecting lines and circles and so satisfies at worst a quadratic over what came before. A transcendental number satisfies no polynomial at all. Therefore pi is not constructible, and the circle cannot be squared. The proof is not that nobody has managed it. It is that the target lies outside the set of things the tools can reach.
This deserves stating plainly, because sacred-geometry writing often treats squaring the circle as a standing mystery or an unfinished quest. As a symbol of the union of heaven and earth, the circle set against the square, the image is a legitimate and ancient one. As a mathematical problem it was answered in 1882, and the answer is no. Good approximations remain useful in practice, and purported exact solutions have continued to arrive in mathematics departments for well over a century.