The regular nonagon cannot be constructed with compass and straightedge. The reason is a detail in the governing theorem that is easy to overlook and that does all the work here.
The Gauss-Wantzel theorem states that a regular n-gon is constructible if and only if n is a power of 2 multiplied by a product of distinct Fermat primes. Fermat primes are those of the form 2 to the power 2 to the power k, plus one; only 3, 5, 17, 257 and 65537 are known. Nine is 3 times 3. Three is a Fermat prime, which is why the equilateral triangle is constructible, but nine uses it twice, and the theorem does not permit repetition. That single word, distinct, is why the nonagon fails while the hexagon, 2 times 3, and the 15-gon, 3 times 5, both succeed.
The underlying reason for the restriction is algebraic. Each new point in a classical construction comes from intersecting lines and circles, which solves at worst a quadratic, so every constructible length has degree a power of 2 over the rationals. A repeated prime factor forces an irreducible cubic into the field extension, and 3 is not a power of 2.
This is a proof, not a gap in knowledge. Gauss gave the sufficient direction in 1796 and published it in 1801; Wantzel proved the necessary direction in 1837. The nonagon has been known to be impossible for nearly two centuries, and no method using only those two instruments will ever produce one. Every nine-sided figure drawn with compass and straightedge, however carefully, is an approximation to something the tools cannot reach.