I hope we can all agree that we cannot prove basically anything in Genesis. There is absolutely no proof of the Garden of Eden, Adam and Eve, etc. Likewise, there is no evidence for Noah/the ark, or any for Moses. But, we know things like the creation of the Second Temple actually happened (there's still part of one of its walls up!). So my question is, what characters in the Old Testament have been proven by non-biblical sources, and what are those sources?
This exact same question lead me to start reading Biblical History and Israel's Past:the changing study of the Bible and history by Megan B. Moore, Brad E.Kelle (from /r/AskHistorians's recommended reading list). I am only halfway through, so I can't give you a final answer on the most up-to-date consensus. However, I can say that the entire book deals with exactly what you're asking (with many additional sources cited), and is quite an amazing read.
Each chapter deals with a different (often chronological) aspect of the biblical story, and doesn't specifically look for "the point from which it becomes accurate". I can try to narrow down your question by giving the "patriarchal age" (the narrative of Genesis 12-50, starting with Abraham, as the "origin story" of the Hebrews) as an example.
Abraham was told to have migrated from Ur in Mesopotamia to Canaan, sometimes around the middle bronze age (that was the assumption of the day). This meant that you could use text from around that time period and location (of which there are plenty) as outside sources (The Nuzi Texts are a notable example). These wouldn't, of course, deal specifically with the 'historical' Abraham, but you could try and draw similarities in the names, traditions and locations used in those text and the Bible to lend more credit to the narrative. The short story here is that while such likeness was found, it didn't allow to pinpoint an exact timeframe for the life of the patriarchs (you would find such similarities across a wide chronological range).
In addition, a close examination of the Bible reveals some anachronisms, for example mentioning cities (e.g Gerar) and peoples (e.g the Philistines) which weren't prominent (or even exist in other sources) until the late bronze age. The appearance of domesticated camels in Canaan is also a point of contention (with current archaeological findings dating their earliest presence in the early iron age), slowly making the consensus shift towards the patriarchal narrative as being a later literary invention which possibly draws from a much older oral tradition.
I hope this serves as a teaser for the book which, once again, I find incredibly fascinating. If you're interested in this topic, you should definitely give it a read.