This is a really tricky subject that is heavily debated. There are some who argue that oral history can go back very far (much farther than written history) and there are those that argue that it only reflects a few decades.

The examples I've been taught include the Homeric epics, modern theory is that the war of Troy happened around 1200 b.c. but the stories weren't written down untill 400-500 years later. So they must have been told as oral history, and this is visible in the text as it rhymes sometimes not in Greek but in linear b Script, so we know that some phrases are centuries older than the writing. But if you look through the text it's also clear that the author does not describe the culture and society of that past but of his own time, and that every once in a while there is something he mentions that he doesn't really understand but that is an important part of the story and thus has to be mentioned.
The Iliad and Odyssey show one problem many scholars attest oral history: it changes, things get swapped out because someone makes a mistake or because they aren't understood any longer in the current society.
There is also an effect noticed in some African cultures that don't have our concept of history that use stories of the past to explain the present, those stories usually claim to be old and unchanged but can change dramatically within a few years as the circumstances change.

All of this is however not to say that oral history is "fake", just that it is streamlined and not always accurate. For example in the Hebrew Bible (which existed as oral history before being written down) there are several examples of events(a prominent example is Passover) that were probably canonised, but relayed somewhat accurately through retelling and rituals. The theory here being that over a few years after the event the eyewitnesses talked about it and formed one cohesive story (instead of many individual ones) and that story is then retold and combined with ritual to pass on to children.

The main problem is that we can't ever know how accurate an old story is, whether it's written or told it's impossible for us to know what really happened back then. (And there are even examples, like Jordans history of the Goth, of supposedly oral history having been written down that may be complete invention or maybe just a little polished and we can probabyl never be sure).

We have had some previous answers that have addressed the way oral history works to record events from the distant past. u/LordHussyPants has a good post here about Maori oral history. u/Snapshot52 discusses using oral history as historical sources in these Monday Methods posts: An Indigenous Approach to History and Indigenous Source: Reconciling Apparent Contradictions.

The question "How far back can oral history go?" depends on how specific you want the information to be. The furthest-back examples of corroborated oral history, as far as I'm aware, come from Australia. Multiple Aboriginal nations have preserved stories of a time when they travelled to coastal places that have been underwater for thousands of years. The Yidiɲɖi people of Fitzroy, for example, have placenames that describe a landscape that last existed in their region 9,960 years ago, requiring an oral transmission of approximately 400 generations. Similarly, the Klamath people of Oregon preserved in oral history a memory of the eruption of Mount Mazama, a former volcano that created Crater Lake, 7,600 years ago. For comparison, these correspond to the Mesolithic and Neolithic periods in Egypt, respectively, thousands of years before the Early Dynastic Period. These accounts date back to before the invention of writing in any society.

These are incredible examples of oral history recounting significant changes to the landscape thousands of years ago. The closer you get to the present, the more specific a lot of corroborated oral history tends to be. For example, some scholars have used the detail of a possible solar eclipse in the story of the founding of the Haudenosaunee Confederacy to date it to 1142. Indigenous generation counts put the foundation somewhere between 900 and 1000, which is pretty comparable. The Pueblo peoples have oral histories about Chaco Canyon being abandoned because its inhabitants were out of balance with the spirits of the natural world. Archaeological evidence suggests that the canyon's massive complex of buildings was abandoned in the 12th century in large part due to climactic instability caused by the inhabitants' deforestation of the Chaska Mountains to get the timber to support their monumental construction projects. These are two examples of oral history being corroborated by outside sources which date to a period equivalent to the High Middle Ages in Europe.

I'm sure people could provide some more examples of detailed and specific oral history from more recent times, but since you asked about just how far back oral history knowledge can go, I hope this gives you an idea.