That's true, but it's important to remember that the amount of waste produced by nuclear power generation is many, many orders of magnitude smaller than fossil fuels.
Let's say you want to generate 1 megawatt-hour of electricity (roughly enough to power an average American household for a month). With coal power, you get roughly 2,000 pounds of carbon dioxide dumped into the atmosphere. [1] To generate the same amount at a nuclear plant, the waste primarily consists of about 3 grams of spent fuel. [2][3]
(And before anyone jumps on me, I'm not trying to dismiss other renewable options... just trying to put nuclear power in perspective.)
That's probably based upon older reactor designs like everything in the nuclear industry, since regulation had made progress slow-going.
Newer designs, and some others, burn spent fuel. Out of the 3 major nuclear reactor incidents, all were old reactor designs. There are new ones that also cannot meltdown in the common understanding of the term.
Don't forget coal mining releases nuclear radiation, since you are unearthing radioactive minerals. So much so, that you are exposed to more radiation in the vicinity of a coal plant than a nuclear reactor.
Now multiply that by the number of households, and again by twelve to see the amount you'd need to permanently store each year.