Not exactly an answer to the question, but perhaps useful in getting a vague sense of the extent of modern European record keeping... In England, anything before 1189 is legally considered "Time immemorial." (The definition has been tinkered with a bit since it was originally established.) The idea was that a person in 1300 might not have access to any documented title to their land which was bought/given/conquered centuries prior. But it was considered theirs if they could establish the ownership as far back as the start of the reign of Richard I in 1189. Any claims predating 1189 were considered too far back to be relevant. What's interesting is that 1189 was picked as the cutoff in 1275, putting "Time immemorial" less than a century in the past at that time.

Europe as a whole has had many legal systems over the past few millennia , so that date wouldn't apply to every part of what was once Roman imperial controlled land. But I would expect that most "ancient" titles would be in the form of "person had this land since the current legal entity's oldest documents" rather than having a paper trail back all the way to Roman times. There were lots of land acquisitions by what we'd now consider extralegal means over that length of time. Though it wouldn't surprise me if there are a few Italians who make the claim of lineage back to Rome for the benefit of the tourists, I don't know how it would be more than just an unsupported assertion at this point.

I initially was replying to a comment that was deleted so I'll just posy this here as it still contains some relevant info...

Veteran's colonies were definitely still a thing as late as Hadrian's reign, definitely much later than the Gracchi. It has been argued that the primary reason for no longer granting land was actually that soldiers preferred to settle in the region of their service and many abandoned their gifted land to return to that region.

It is quite interesting to note that in the west veteran colonies, whether from settlement or these aforementioned soldiers retiring in the region of their service, became one of the main sources of new troops. So there developed almost a tradition of military service, with ex soldiers sons being one of the main sources of new troops as late as the 4th century.

I couldn't comment on whether there are still their descendants there today but with whole towns developing over centuries this way, it could be way more probable than you think.

Source

Mann, The Settlement of Veterans in the Roman Empire

I have a bit of an answer for you, but it'll need to be filled out by a proper historian who knows the fate of the ager publicus post-antiquity.

While I cannot say if there are any particular plots whose deeds were continuously passed down from antiquity within the same family, I can at least give a basic rundown of how public landholding tended to work, and that should at least give you an idea how likely it is that any such titles may have survived uninterrupted.

Sources for all of the following include Cicero, Tacitus, and Livy.

Part of the 'rights' of being a Roman Citizen was the promise of land, most especially land through conquest. There was really nothing more noble or 'Roman', to the ancient Roman mind, than good, honest farming, so this land was highly prized. This ager publicus (I believe it means 'public farmland') was parceled out to individual Romans, usually post military conquest. This was to be done in amounts no more than 500 iugera (rather like an acre) per person. The parcel size limit policy was oftentimes ignored or circumvented, though, and Aristocrats and Equestrians frequently received more than their fair share. The law had been on the books for a long time in Roman history, but it wasn't until a few centuries later that anybody seemed to pay attention to it. The policy of land distribution in general existed since the early days of the republic, and I believe continued up through at least the early Imperial era - so the following I believe is accurate through at least the third century.

Initially, this awarding of land wasn't done to military men, at least not specifically. Military men in the early through late republic already had land - in fact it was part of the requirement to even join and serve. Good, honorable romans had farmland, and you didn't want any of those landless scalliwags out there fighting for money or land when they should be fighting for Roman ideals. This notion didn't last forever, though. Post Marian reforms it became necessary to award land to troops, like you mentioned, partly because the Roman army was no longer exclusively composed of landholders. Marius, for reasons I won't get into, restructured the army such that almost anyone could join - landholding was no longer a requirement. Thus, Roman generals worked very hard to secure good farmland for their troops, so their men (often not formerly 'good, honest Romans' because they had no farmland to their name) could ascend the social ladder. This in turn secured loyalty for the troops to their General, which could be for good or for ill, depending on how you look at it. In the view of many who've studied Roman history, land distribution to troops was a leading cause to the fall of the republic.

Getting more to the point, as for what often happened to such lands, it was Roman custom to divide the land equally among all male children, and the common landholding citizen tended to have several. The aristocracy tended to have few, (which will be important later) so a common problem -which the middle classes were acutely aware of- was the ever diminishing land a family would hold. Men divided their lands among their sons, who divided it further among theirs, and so on. This meant that eventually a given non-wealthy inheritee would often find that his endowed acreage was too small to realistically earn a living by. He would often sell that little parcel of land to someone who could afford to buy it and seek his fortunes elsewhere- and this was usually an Aristocrat or someone from the Equestrian order.

As such the upper classes tended to accumulate more and more land, being given more than their share initially when new public land allotments were declared, and scooped up even more by purchase when the opportunities presented themselves. Additionally, your typical upper-class Roman man was having very few legitimate children, and so the children he had could be reasonably confident that he would eventually inherit a hefty acreage that generated enough income to grow by purchasing smaller plots the lower classes were seeking to sell.

This accumulation over time led to many attempts to redistribute land and enforce the 500 iugera limit, but most attempts met with little lasting success and a great many ended in bloody massacres of the reformers pushing for the redistribution. This cry for reform, resistance by the aristocracy leading to riots for reform, which led to a massacre of the reformers happened almost cyclically for a couple centuries to the point of being fairly predictable. As such, I think the likelihood of a plot of family farmland surviving uninterrupted from antiquity is fairly small, as the farmers small enough to avoid notice from the events of history tended to eventually sell it to wealthier folk, who were often targets of events themselves because they were so wealthy.

Follow-up question to the one asked: What is the oldest still-existing title still honored by a government?

While I doubt there are anything as specific as land titles, Roman centuriation is still visible in many areas of Europe, such as Padua, around Naples and southwest England (to mention only some areas I have familiarity). So the land use patterns still have an effect on property divisions.

According to Cassius Dio, Augustus abolished land grants to Roman soldiers in favor of fixed monetary payments because the Roman government was not always able to fairly fulfill their promise of land. Roman soldiers and veterans clamored about their grants to such a degree that Augustus was concerned about a revolt.

Taking this Augustine reform into account, any land grant documentation would likely be from the Republican era and be over 2,000 years old. Considering the sheer age and the number of governmental transitions, survival of these documents would be unlikely, and the idea of any government or ruling authority post-Roman Empire actually honoring these documents seems even less likely.

Later land grants would likely trace back to the Latifundia or living space provided to the Foederati on the Empire's borders. In any case, to be able to find a land title/deed from any of these eras would probably be considered an exceptional find. Being able to tie that same land title/deed to a current genealogy would be just short of impossible.