At least in the theoretical sense, this idea is reasonable. If the carousel were spinning so that the outer edge of the carousel moved at nearly the speed of light, then time would appear to slow down for people on the carousel. When the carousel riders looked out at the world spinning by, the days would pass very quickly. So the people on the carousel would age very slowly relative to people not on the carousel. This would create, essentially, a time machine that lets the passengers on the carousel travel into the future. See How Special Relativity Works for details.
In a practical sense, this idea has problems because of the centrifugal forces that the carousel would generate. Some of the fastest rotating objects in existence today are high-speed flywheels. High-speed flywheels float on magnetic bearings in a vacuum chamber so there is very nearly zero friction on them. These flywheels are able to achieve speeds up to about 200,000 rotations per minute (rpm). The main problem with flywheels running this quickly comes in the form of rotor disintegration -- the outward forces on the rotor are huge. Even if you assume that a flywheel with a 3-foot (1-meter) circumference and about a 12-inch (32-cm) diameter were running at 1 million rpm, the outer edge of the flywheel would be traveling at only about 34,000 miles per hour (55,000 kph) -- nowhere even close to the speed of light at 186,000 miles per second.
The other option would be to create an extremely large carousel. Say you built a ring as big as Earth's distance from the sun, or about 186,000,000 miles (300,000,000 km) in diameter. Its circumference would be about 584,000,000 miles (942,000,000 km). This ring would only have to spin at a rate of about one revolution per hour (compared to the Earth's one revolution per year) to achieve something near the speed of light. Building this ring and then accelerating the entire thing, of course, would be a big project...
To create a time machine like the one you are describing, you would really have to do it with a spacecraft moving in a straight line through space. The main problem then becomes accelerating the spacecraft. Let's say that you want to accelerate to a speed approaching that of the speed of light, and you are willing to subject the passengers to 2 Gs of force (twice the force of gravity -- people weighing 150 pounds would feel like they weigh 300 pounds) during the acceleration process. The passengers would have to endure that force for about half a year! Never mind the amount of fuel it would take to provide that acceleration...
In other words, no one is going to be traveling near the speed of light anytime soon.
These links will help you learn more:
- How Special Relativity Works
- How Time Travel Will Work
- How Light Works
- How Rocket Engines Work
- Living in a 2-G World
- Patent #5,124,605: Flywheel-based energy storage methods and apparatus - good overview of high-speed flywheel technology