Nuclear power plants produce very little waste. According to the Nuclear Energy Institute, all the nuclear waste produced by commercial plants since the late 1950s only occupy the space of a football field about 9 meters deep.
By stark contrast, the coal industry produces the same volume of waste each hour.
While there may not be a perfect solution for nuclear waste, most scientists agree that burying it deep underground would be the safest possible option. It's just a matter of where. The U.S. has been trying to open up permanent disposal sites for decades, like Yucca mountain, but they have constantly been thwarted by politics.