As others have said, the scale of both the construction and the forces involved place this project firmly in the realm of the impossible. However, that does not mean the idea hasn't been considered in a different form.
While not exactly a catapult, There has been a somewhat similar idea involving what's basically a particle accelerator for space cargo that can withstand the pressure. Imagine the Cern Large Hadron collider, but built into a mountain with an exit port aimed at a stable orbit entry point. The theory is that you get the craft up to speed on the ground floor and then launch it into orbit from there. While the G-forces are not viable for anything living, properly protected equipment that can withstand both the G-forces and the magnetic forces involved could be brought into outer space for a ludicrously low price compared to traditional rocket-aided launch methods (on the order of 10,000 times lower for the launch itself).
sources:
- Wikipedia article on Mass Drivers: http://en.wikipedia.org/wiki/Mass_driver
- Wikipedia article on Space Gun: http://en.wikipedia.org/wiki/Space_gun
- Article on New Scientist: http://www.newscientist.com/article/dn10180-huge-launch-ring-to-fling-satellites-into-orbit.html
Also see the different sources on the Wikipedia articles.