It seems natural to me to use the expression "without barely X" as a translation from Spanish of "sin apenas X". But I have found very few instances on Google, which seem to be mostly translations. "With barely X" seems slightly more common, but not that much. Do they sound sound natural or are they "correct"?
"With barely X" could work, but "with hardly X" is better (actually, Collins' Spanish-English dictionary has a lot of examples of usage that could help you in many cases—it's an excellent dictionary).
"Without barely" does not make much sense, because "barely" indicates a slight presence of something, while "without" is about an absence of something (um, not sure it's helping you because that same logic could be applied to Spanish, but there, "sin apenas" actually works). Anyway, when you want to say that there was not even a hint of X, the correct way would be to say "without so much as".