The nonagon's impossibility has a familiar face. A regular nonagon has a central angle of 40 degrees. An equilateral triangle, easily constructed, has a central angle of 120. To get from the one to the other is to divide 120 degrees into three, which is to trisect an angle.
Trisecting an arbitrary angle is one of the three classical problems of Greek geometry, alongside doubling the cube and squaring the circle. It is impossible with compass and straightedge, and it was Wantzel's paper of 1837 that proved it, the same paper that settled the constructible polygons. The nonagon and the trisection are two statements of a single fact.
Note the qualification: an arbitrary angle. Plenty of particular angles trisect perfectly well. A right angle divides into three 30 degree parts without difficulty, because 30 degrees is constructible. What fails is a general method, and 120 degrees is a concrete instance of that failure.
Approximations are entirely serviceable in practice. Durer's Underweysung der Messung of 1525 gives one, using three lens-shaped vesica figures set on the vertices of an inscribed equilateral triangle together with a division of a radius into thirds. Note the distinction that makes: dividing a straight segment into three equal parts is elementary, and Euclid does it; it is dividing an angle into three that cannot be done. Durer's nonagon is accurate to well within the tolerance of any craft that would have used it.
Exact nonagons become available once the rules are relaxed. A marked ruler used in neusis will trisect an angle, as Archimedes knew, and so will paper folding, and so will several classical curves invented for the purpose, among them the conchoid of Nicomedes and the quadratrix. The nonagon is not beyond geometry. It is beyond two particular instruments, and the theorem states exactly why.