The answer to this question is quite complex and depends not only on the historian, but also what particular bit of history we are talking about. Between the things everyone agrees are "real history" (eg the existence of Julius Caesar) and things everyone agrees are mythical (eg Romulus) there is a lot of grey area. For example, the Greek historian Polybius recorded the text of several treaties between Rome and Carthage, the oldest from about 500 BCE. As far as I am aware there is not much debate on whether or not these are real it is broadly agreed that Polybius was not simply lying. This treaty gives a pretty interesting and revealing picture of Rome's position in the western Mediterranean from an early point, but it was also more or less contemporaneous with the story of the Rape of Lucretia and the expulsion of the Tarquins by Brutus, and the subsequent war with Lars Porsenna, which are generally agreed to have been, at the very least, heavily mythologized. So from one perspective we can say we have important and accurate information about Rome in 500, but from another we can say that period is essentially mythical. Adding archaeology into this mix only adds further wrinkles.

There are even less exaggerated examples, to give one, everyone agrees that the First Punic War happened, and everyone agrees on the broad outline of its course (very broad, unfortunately for such an important conflict we have rather scanty information), but there are quite a few details people wrangle over, from the question of whether the corvus was an actual thing, to the story about the Romans building their navy by reverse engineering a crashed Carthaginian ship (which I think is utter nonsense but it gets repeated), to the black hole that is our understanding of the Carthaginian side of things. And then things like Caesar's Gallic campaigns being pretty well understood, but were there really several hundred thousand Gallic warriors at Alesia? Feels kind of dubious.

Beyond that, a few rapid fire dates that are often given for this sort of "start of real Roman history":

390 BCE: the sack of Rome by the Gauls supposedly destroyed the old records of the city, urning events prior into dressed up myths.

312 BCE: censorship of Appius Claudius, and epigraphically attested figure who gets arguably the first really human portrait in Roman history.

290 BCE: the Pyrrhic War, when Rome entered into the Greek political environment.

264 BCE: Start of the First Punic War, but more importantly when the surviving text of Polybius, generally considered the most reliable early historian of Rome, picks up.

So pick your poison.

A good source for this is TJ Cornell's Origins of Rome, he is generally unusually credulous of early sources but in building his case he really drills down into the sources' sources' sources, so to speak.