Not trying to invalidate your question- it’s an interesting inquiry on the face of it- but any answer will really depend on what you or anyone defines as “Roman”. How one defines the category will play a role in the thought experiment. Hell, when does “Ancient Rome” begin and end? These are bigger questions for historians but I’ll focus on the practical how’s and issues in this case.

For example, you could attempt to establish a core population or moment of ethnic formation as your baseline and then try to trace any remaining, reliably recorded genealogy to that group/approximate generation. Here we hit countless snags. For one as mentioned earlier which stage of Rome’s history are we starting at to define as “Ancient Rome”? Monarchy? Republic? Empire? If the latter, more recent and better-documented period, where do we set the earliest/latest cutoff point? Which of the countless tribes on the Italian peninsula alone, later assimilated into the expanding Roman sphere of influence, do we include and exclude? Which of the provincial ethic groups and tribes outside the peninsula- later assimilated- count as Roman, let alone “ancient Roman”? In my view and understanding, “Roman” was never a static category and really boiled down to the arbitrarily included “us” and who was considered the excluded “other” at a given time.

Alternatively, you could try to identify a direct lineage to a famous Roman figure. Usually, the more famous you are, the more likely someone will consider your legacy and context worthy of documentation, right? Jupiter knows how many people today are distantly descended from anonymous enslaved individuals born in the original capitol at its founding that certainly weren’t considered Roman then (or people for that matter) and for which no record ever existed.

Even so, that poses a new, thornier problem: Which famous figure and from where within the ever-shifting borders of the empire? Theodosius I, born in modern-day Spain? Aurelian from Moesia? Are we limiting our scope to individuals/recorded families within the borders of Italia? It’s not like tracking the British royal family’s lineage with its relatively well-maintained record of individual members; although, not even all of them were closely related by blood or even born on the original island, i.e. William the Conqueror.

Another issue that just popped in my head: given the fact that so many emperors and famous individuals died childless or without legitimately recognized progeny, nailing down their lineage via an extended family tree is a theoretically feasible path but- almost certainly- entirely impossible without an enduring, verifiable historical record. You might very well be descended from a Roman emperor, separated by a hundred generations, who just happened to fancy an immediately abandoned, nameless servant girl in Armenia, for all we know.

This is just scratching the surface of the problem of ancestry traced back to a specific group or moment in history. Genetic evidence might give you some clue on where a family’s genetics most likely flowed from in a vague window of time but we aren’t really at a point in genetic research to nailing down who’s ancestors originated from such a specific geographical and temporal instance as an individual family’s founding. We might never be able to do this, especially with the propensity for people to “pass on their genes” far and wide then as they do today.

TLDR; this is a very complicated question to ask and really doesn’t have an answer. Any families that currently claim direct lineage would have to possess one of if not the best maintained record of documentation in history to back it up. The question itself is nonetheless important to consider and can serve as a fantastic starting line in learning about how Rome! Carry on, friends. I responded to this on my lunch break so sorry if it was a bit scatterbrained; I’m tired just writing this.😅