Yep, every kid had one, and if you didn't then you weren't cool
Jokes aside, child chemistry sets came with radioactive elements back then. (in real life not the game world). Everyone was obsessed with nuclear technology and radioactive materials, so yes they put cancer causing radioactive material in a child's toy. And yes, many people got cancer as a result.
Also, if you pay attention to the subtle details in Fallout 3 a lot of companies/products were named after nuclear stuff. My favorite is the home appliance company called Radiation King that made ovens and refrigerators and stuff.
If you pay careful attention to the story elements of the game, such as old newspaper articles on computer terminals and the diaries of people who lived before the Great War, you will notice that the world knew nuclear war was coming for decades before it happened. Similar to how everyone in the 50's thought Mutually Assured Destruction would happen. Except in the case of Fallout, it actually did.