The post that was quoted here has been removed
I take Rand's statement to mean that contradictions don't manifest in reality. That is, it is never the case in any actual universe that both A and Not-A are true, where A is any claim about the way the universe is.
I don't take Rand's statement to mean that logical statements representing contradictions can't be constructed or conceived of. On the contrary, counterexamples to such an assertion are trivial to find. Here's one: the proposition (A and Not-A).
This interpretation applies to Russell's paradox in two main ways.
First, Russel's R, the set of all sets that are not members of themselves, can never manifest in reality, the domain to which I take Rand's claim to refer, so R can never both be and not be a member of itself, as it can never be at all. Thus, R is not a counterexample to Rand's claim. I believe Hilbert's analysis of the R paradox is along these lines.
Alternatively, Rand's admonition to check ones premises upon finding a contradiction is precisely what set theorists pursued, an endeavor which has enriched the field and led to better understanding. Contrast this to rejecting the admonition, in which case set theorists would have just thrown up their hands, said "Well, that's weird" and moved on. The prior axioms simply corresponded to a set of premises that obscurely entailed (A and Not-A). I think it is clear that the possibility of constructing such a set of premises is not what Rand was denying.
Finally, I think it is generally accepted that Rand was a legitimate child.