Steelmaking is the combination of iron ore and carbon (from coked coal) with huge amounts of forced air or direct oxygen to form the alloy of carbon and iron we call steel.

One notable form of radioactive contamination is Carbon-14, which is what makes radiocarbon dating after ~1950 unreliable. Though of course since the carbon in coal is itself primordial, that isn't the principle route of steel contamination.

Best I can make out it's radioactive isotopes in the air itself which increase the radiation background of post-WWII steel, with several sites mentioning Cobalt-60. Substances used in the post-smelting processing of steel (welding rods and the like) may also introduce contamination.

Given the 1963 Partial Test Ban Treaty which has halted most atmostpheric nuclear testing, radiation levels have fallen to the point that current steel is largely similar to pre-WWII "low-emissivity" steel in terms of background radiation.

<https://en.wikipedia.org/wiki/Low-background_steel>

When you turn iron ore into steel with blast furnaces and steel making (e.g. basic oxygen process) you blast it with atmospheric air (or oxygen made from atmospheric air), and tiny amounts of impurities in the air (such as fallout from atmospheric nuclear testing) get embedded in the steel.

Yes, this is the same air you're breathing. No, the levels are not high enough to be a health issue. But it's an issue for very sensitive scientific instruments and such.

It's because the smelting process introduces a lot of isotopes into steel when you make it.

I assume that they find big pieces of already smelted steel in the ocean and just grind away the rust / mill it into the shape they want.

mostly because steel is used for things like particle accelerators that are understandably very sensitive to contamination - as a sibling comment noted this is not much of an issue anymore.