Nuclear waste is a blanket term for a large amount of radioactive things. Not all nuclear waste is created equal, they are often classified into three levels. The first level of waste comes in the form of lightly irradiated things, like hazmat suits and tools that came in close proximity to radioactive materials. This tier constitutes to 90% of total nuclear waste and thankfully so. It is also known as low level waste or LLW and requires no shieldingand is disposed in a normal area such as a landfill.
The second tier of nuclear waste are parts that have been in close proximity or even touching the fuel rods. It also includes radioactive sludge, metal cladding for nuclear fuel. This tier constitutes to 7% of total nuclear waste and requires shielding for safety and is often stored in geological repositories, shallow ones for the short term radioactivity and deep ones for the long term radioactivity. It is also known as intermediate level waste of ILW
The third, and most dangerous tier is the spent fuel rods, fresh out of the nuclear reactor and ready to eat.* It can also come from nuclear fuel reprocessing, a process where spent fuel rods are repurposed into new fuel rods. This tier constitutes to 3% of total nuclear waste (thankfully). Spent fuel is highly radioactive and hot and can spontaneously explode. This waste is melted and mixed with glass, stored inside large concrete caskets, and is shipped to somewhere to be safely contained. In the United States, we store all of our HLW (high level waste) in a place called the WIPP or the Waste Isolation Pilot Plant. It is a massive underground maze like complex designed to store nuclear waste for a very, very, very long time. There were plans to make a spike forest around the complex to warn future civilizations to not go near our waste.
Overall, unless you are a trained professional, do not mess with radioactive material.
Related Articles
https://world-nuclear.org/nuclear-essentials/what-is-nuclear-waste-and-what-do-we-do-with-it.aspx
https://www.nrc.gov/reading-rm/doc-collections/fact-sheets/radwaste.html
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*DO NOT eat spent nuclear fuel, it is dangerous and you would be dead within the hour