There’s a problem in the premise of the question. An object does not itself become radioactive after being exposed to radiation. Just like an object exposed to sunlight will not itself become a source of light (neutron activation is an exception, but in most cases is not relevant).
Radioactive contamination is a different matter. In this case, an object is exposed to eg. dust, gas or aerosol of a radionuclide, like Cs137 or I131. In this case, the contaminated object can simply be thoroughly washed, removing the contaminant. This however is sometimes not easily done, depending on the object and the type of contamination. For example, if a person is internally contaminated by Sr90, it will tend to bioaccumulate in the bones, because the chemical properties of strontium are similar to those of calcium. Obviously, decontaminating a living person’s bones is not a simple task.
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