Fermat’s Last Theorem is the most famous modern (since 1600) long-unsolved problem. He actually did write that in the margin of a book.
It was finally solved a few years ago.
Another modern problem was to find the roots of the general fifth-degree or higher polynomial by basic algebraic operations and taking roots. It was shown impossible in the 19th century.
Famous ancient problems would be trisecting the angle and duplicating the cube by compass and straight-edge, which were proved to be impossible in the 19th century.
Another ancient problem was to construct the regular septagon with compass and straight-edge, which Gauss did.
It's the regular heptadecagon, i.e. 17 sides. The heptagon, or septagon, is inconstructible, with 7 sides, as is the nonagon (9).
Also Gauss didn't actually devise a construction. He just proved it was possible as the length a side in terms of the radius of circumscribing circle is "solvable in radicals". Actually, it's very interesting, and I've been studying it. You can quickly find animated versions of actual constructions on the internet.
I would say that you can make a better case that the Riemann Hypothesis [still unsolved] is more famous, and it is certainly more important in terms of its theoretical and practical implications. It doesn't go back as far. Goldbach's conjecture is also not quite as old, but just as famous [until Andrew Wiles proved the Fermat Conjecture it wasn't that well known to lay audiences -- it was not one of the Hilbert Problems; the Riemann Hypothesis is.]