Why is it impossible to cut out a 32 unit square?

Answer 1

It is not! All you need is a square whose sides are sqrt(32) = 4*sqrt(2) in length.

You might try and argue that 4*sqrt(2) is an irrational number and so cannot be calculated exactly so it is not possible to draw a side of length 4*sqrt(2). Faulty argument.

Draw a 4*4 square. Easy enough! The diagonal of this square has length sqrt(32) = 4*sqrt(2). So now you have a line of the required length.