This happened when I was slicing a cylindric piece of sausage. Tired of round slices, I put the piece upright and noticed that the slices are now rectangles of different sizes. Then, I transfered this to the circle as something like an “infinitely thin piece of sausage”. Cutting it in “upright position” (sounds strange with height=0, though) and cutting inifinitely thin slices gives rectangles of height zero, that is, lines. Now, you can connect the lines piece by piece to one infinitely long line, and cut that into an infinite number of other lines, each the lenght of one edge of the desired square. Now put the lines next to each other like matches in a box (but only one layer) and you have the square.
I looked the problem up in Wikipedia afterwards, and solutions like this one are probably meant where they state: “Bending the rules by allowing an infinite number of compass-and-straightedge operations […] also makes squaring the circle possible.” (English Wikipedia, on the impossibility of squaring the circle).