According to the book The Irish In the South, 1815-1877 by David T. Gleeson:
"The Irish supported African slavery in the South. Wealthier Irish men had no qualms about buying slaves, and those too poor to puchase chattel did not oppose the institution."
Gleeson points to a couple of the wealthiest Irish immigrant slaveholders. Frederick Stanton, who immigrated to Natchez, Mississippi, owned sixteen plantations in Mississippi and Louisiana by the time of his death in 1859, and held 333 black people in slavery. Another Irish immigrant, Maunsel White, owned four plantations in Louisiana by 1850, one of which held 192 black people in slavery.
But as Gleeson explains:
"Like most southern slaveowners, however, the majority of Irish slaveholders did not reach the heights of Stanton or White. Many had to work beside their slaves. The Irish settlers of Locust Grove, Georgia, and Sulphur Springs, Mississippi, had only a few slaves on their farms. John Ward owned five slaves, including two children, while his neighbors Patrick Burk[e] and John Kelley owned just one slave each. One of the original Irish residents of Locust Grove, Murtha Griffin, did manage to acquire forty-two slaves by 1860 despite increasing prices."
So, uncommon? Yes, but white people who were part of the plantation-owning aristocracy were uncommon in general. But impossible, or anachronistic? No. It did happen.
One other thing to keep in mind is the ancestry outlined in the book. While I haven't read it myself (so someone correct me if I am wrong), only Scarlett O'Hara's father is Irish. Her mother Ellen is American-born of French descent. Scarlett's maternal grandfather is named Pierre Robillard, whose family is Catholic, but he has converted to Presbyterianism at some earlier date. The family also have relatives living in Charleston, South Carolina.
It might be slightly more believable if they were living in Louisiana instead of Georgia, with Scarlett being the offspring of a Cajun mother and an Irish immigrant father to New Orleans. And since the backstory of her parents, as far as I know, is not fleshed out in that way, it could be assumed that something like this is what happened. But even if her mother grew up in Georgia and her father immigrated there, it's not completely unbelievable. The Catholic Diocese of Savannah was established in 1850, and covered Atlanta and the rest of the state. So it's not like there weren't Catholics living there. And Charleston, where they had relatives, had established its Roman Catholic Diocese in 1820, and had a longer history of Catholic residents. So, whether Scarlett had Cajun roots, or South Carolina roots, or simply Georgia roots on her mother's side, the plot of a mixed Irish-French Catholic family being wealthy Southern plantation owners and slaveholders is entirely possible.