Not that it matters, but it's perhaps worth pointing out that it is rather badly written. For the following reason:
The phrase slang "did time" is slang very much from the UK or US "gangster" era; a typical dictionary definition example will be from Mickey Spillane or the like.
In stark contrast, "Siberia" relates to issues like gulags, concentration camps, political prisoner camps, death marches, the Soviet Era, Aleksandr Solzhenitsyn, and other depressing aspects of Russian literature and history.
Note that quite simply you would, indeed, never say you "did time" in a Gulag. (!)
Indeed, perhaps an even more unsavory phrase, you would certainly never say you "did time" in a concentration camp. (!!!)
To make perhaps an analogy, Shafizadeh, you would not take slang from the "rap world" of Eminem and so on, and kind of mix it with slang from (say) the "disco!" era. it would be sort of silly right? The "mixed slang" here is rather groan-inspiring. I "did time" in a "gulag". Yeah.
So, yes, "did time" is simply UK/US slang for "was in prison", it's "gangster" slang (think of say 1950s gangsters). In a rather conjunction, "Siberia" relates to "prison camps" and "gulags".
So for example, a far more sensible phrase would be something like, I did time in some horrible prison, say, "my heart did time in san quentin" sort of thing.