Greek and barbarian were stereotypical labels. People who were cultured citizens of the Roman empire could be called Greeks; the rest were barbarians.
"Barbarian" was an onomatopoetic word, which means that it sounds like what it represents. They were called "barbarians" because their language sounded like gibberish (bar-bar-bar-bar). This would be like you or me labeling people Gobbledygookians because their speech sounded like gobbledygook.
So when Paul says that he has a gospel-debt to both Greeks and Barbarians, he means that he has a mission to all strata of society--both the high and the low, the cultured and the uncultured, the rich and the poor, the cool and the uncool, the urban and the rural, the wise and the foolish.
The white collar and the redneck.
Not many West Virginians are rednecks as the world thinks of rednecks. But we are frequently and prejudicially labeled that way. Most of us are in fact blue collar, and quite a few are white collar. But we have an unusual reputation for redneckedness, even if we are not really all that different from the rest of Appalachia. Many of us embrace that, because that reputation is the only thing that makes us different. You have to hang on to whatever you can when you have nothing. But our problems exist everywhere in this nation; they just seem to exist in greater concentrations here for some reason. Maybe that is what makes us different.
Regardless, I do not think it is a stretch to suggest that Paul includes us when he says in his own way that "he is obligated both to the white collar and the redneck."
Whether we are white collar, blue collar, or redneck, I see us fitting into Paul's spectrum of gospel-debt obligation. Which means that, in spite of "cool-and-cultured" ministry trends, he would not forget about the barbarians. If Paul were alive today, I think he would be praying that somehow, by God's will, he might at last succeed in coming to West Virginia.