I would not have thought it possible, be here we see the photo proof that a C-130 can in fact land and take off from an aircraft carrier. This occurred on the USS Forrestal in 1963. It is to this day, the heaviest and largest aircraft landing and takeoff from a carrier. Amazingly, this USMC KC-130F, with a Navy Lieutenant at the helm, landed on the deck without the use of the arrester cables. The C-130 can change the direction of its propeller blades in order to slow the aircraft and land in a much shorter strip. The crew did this just prior to landing to decrease the space needed to stop the aircraft. Records state that the Hercules made 29 touch and go landings, 21 full stop, unarrested landings and 21 unassisted (no steam catapult) take-offs. Landings and takeoffs were made at several weights from 85,000lbs to 121,00lbs. When at low weight, the Hercules was able to stop in 267 feet, only slightly more than double the wingspan! At the max tested weight, it only required 775 feet for takeoff and 460 feet for landing. This very aircraft was in fact, in service up until 2003 and is now on display at the National Museum of Naval Aviation in Pensacola, Florida. The Navy ultimately decided that the margin for error was too small for operational use and developed the C-2 Greyhound for the COD (Carrier On board Delivery) mission.
For those that might doubt the veracity of my reporting on this matter, there is a Youtube video of a landing, but embedding was disabled so you will have to go to Youtube all by yourself to see it.
Oooh look! Color photo of takeoff!