Romans did not have surnames in the way we have them, but a nomen gentile that signified belonging to a certain gens or extended family, of which specific branches could be identified by another third name, the cognomen, like the famous Caesar branch of the Julii, or the Scipio branch of the Cornelii. See also how did Roman names work?

In short this meant that all Roman citizens usually had two (female) or three (male) names for most of the time. This system broke down over the 3rd and 4th century for various reasons, and singular names became the norm again. Many of these names were new, christian creations relating to christian mythology (like Quodvultdeus, Exsuperius, Refrigerius, Anastasius; some biblical names were also prominent, especially Johannes and Maria); but many others evolved out of former cognomina, and some also from the nomen gentile. In that way, many names of gentes (or 'surnames', if you will) survive in modern onomastic material - examples would be Julius/Julia; Cornelius/Cornelia; Claudius/Claudia; Sergei (from Sergia); Emilio (from Aemilius); Fabio (from Fabius); Titus; same goes for personal names such as Marcus and Sextus/Sixtus and so on, but generally not as family names but as first names - the trend to hereditary additional names to the single personal name only began to pick up again during the Middle Ages, and most modern family names also only came into being in modern times.

I have written about this before, if you want the background as well as the relevant scholarly citation.

However, in the interest of keeping things fresh, I'm going to re-emphasize a perspective I wrote in that post.

Surnames are status symbols which are highly attached to the socio-economic system that spawns them.

A (prestigious) surname only makes sense if you have a socio-economic circumstance that favors identifying you with the antiquity of a particular blood lineage. If you don't have the socio-economic circumstance of that, like say if you existed in a purely meritocratic social world with no favoritism toward dynasty, or one consisting of a village where everyone knew each other personally, surnames aren't necessary. Instead, as mentioned in that previous post, descriptor surnames are sufficient. John the Carpenter. William the Smith. Thomas the Tank Engine.

Technically, the way our surnames function now is a hybrid of the lineage / descriptor. Unless your last name is and can be directly linked to an influential dynasty (Rockefellers, Bushes, Clintons), your surname is primarily an additional descriptor to clarify your identifying name among the millions of similar names that now exist because of our complex state structure requiring personal identification.

But if you remove this complex world, and go into Walking Dead mode (which can arguably be a comparison with the early medieval world), surnames cease mattering. Each person in your immediate sphere is now immediately recognizable by a single name, and if you have two Johns, only then do you add an additional qualifier.

Now, back to your question. Have any Roman surnames survived into modern times? The answer is yes, no, and we don't know (God I love history).

Yes because there was an interest during the high medieval period among Italians to claim a certain Roman ancestry of their surnames. The Massimo family is the one that comes immediately to mind. They claim descent from Quintus Fabius Maximus, of the Fabian strategy. So on a superficial basis, "yes" because even the claim of direct descent means those Roman surnames have survived.

However, it's also "no" because we don't actually have any records attesting to a continuity of that name between the late antique and high medieval gap.

Which simultaneously allows us to say "we don't know" since just because we don't have the records, doesn't meant they don't exist. But it doesn't mean they exist either.

tl;dr - the socio-economic world where these surnames existed in antiquity ceased mattering in the early medieval world, so those surnames disappeared. Once the economic circumstances favored claiming descent from antiquity, surnames began re-appearing, but whether those names are continuous to antiquity, is anyone's guess. If I had to bet my money, I'd say no, because it's easier to CLAIM an ancient name than it is to retain one despite changing socio-economic pressures.

And if i may ask a follow up, what is the oldest known surname? can any family trace their linage to the bronze age and beyond?

How does this topic relate to modern Italian last names with Latin-esque spelling? I know families with the last name Di Bernardiis, or Di Fabritiis. How did names with that spelling make it this far? Nothing else in Italian is spelled like that.

This might be of interest OP:

On the evolution of Roman nomina in the post-Roman period

Long story short, Roman names seem to have survived in various states of preservation. It's just that nobody can prove an unbroken chain of descent to a specific family with that name. Naming conventions, society, and language changed too much over the centuries.