(I welcome the corrections or input of a native speaker!) it doesn't seem like the most natural or meaningful sentence, but it would seem to mean that these women (ele) live their lives deeply embedded in a state of corruption, that probably dictates not only their actions but those of the people around them. to be more clear, one could certainly have said, "sistemul lor e corupt" or that "ele sunt corupte" to make it a bit more clear, but I think the authors were creating a simple sentence that would use the noun coruptie, and not the adjective corupt.
I'm also wondering this: can the verb "a trai" be used like an emphatic form of the verb to be? That works in Latin: "vivis molesta" (you LIVE annoying) is a strong way of saying you ARE annoying. can this work the same? So, Ele trăiesc în corupție would then mean "they're very corrupt"?
As an English sentence this is meaningless. No-one could interpret it without further explanation. Yet again this is word for word translation that does not work. What is not clear is whether there is some general rule about how Romanian expresses certain ideas (that an English speaker can understand), or whether this is a specific idiomatic phrase.
In English there was, maybe still is a phrase: "they live in sin". It had / has a specific meaning, of two people living together and having physical relations (trying to be polite / discreet) without being married. FWIW this phrase reminded me, though I haven't actually heard the phrase used for a while now!