Can radiation go through copper

That's a trick question - kind of like "how long is a piece of string?"

In fact, the answer depends on 2 factors: the strength of the radiation source, and the "density" of the matter (in this case, copper) that the radiation is going through. Sort of like "can a flashlight shine through a black cloth?" - it obviously depends on how opaque the cloth is, and if you're using a cheap flashlight or an airport landing lamp.

We think of "density" as the relative weight of a material of a given size piece of it. As such, copper is not as dense as lead (the usual material used to insulate against radiation). But it's a lot more dense than air. However, 5 pounds of air will stop radiation as effectively as 5 pounds of lead, believe it or not.

It's just a lot easier to construct containers and barriers out of lead than out of air (and they're a lot thinner, too).