While the problems have been around for several thousand years, their solutions have not.  In fact they were only definitively proven to be unsolvable in the mid 19th century.  (General methods for determining the solvability of geometric problems were proven in 1837 by Pierre Wantzel.  The last of the 3 problems, circling the square, was laid to rest in 1880 when Lindemann proved that pi was transcendental.)  During their lifetime the classical construction problems have caused more than their fair share of hair pulling.  The ancient Greeks used to refer to someone who attempted the impossible as a "circle squarer", and they even had a word which meant "to busy oneself with the quadrature" referring to the same.  Even in later years the famous mathematician DeMorgan proposed the term "morbus cyclometricus" to mean the circle squaring disease. 

However, along with a lot of sleepless nights the three classical construction problems were also partly responsible for many of mankind's mathematical achievements.  Along with influencing our earliest work in geometry, they also laid the groundwork for ideas that lead to integrals and the more modern theories of groups, rings, and fields.