This source claims, and walks through proof, that the 5 Books of Moses (Pentateuch) date all the way back to at least the era of Joshua ~1200 years B.C.E. Is this really true?
The source you quote makes the case for authorship in the Mosaic era (or even by Moses himself, as is the tradition), based on internal textual evidence and where there is an apparent lack of motive for forgery. This is a reasonable example of one form of evidence for dating the Bible.
A more concrete archaeological example, which you can read on hermeneutics.SE, is the resemblance of the 10 Commandments and Deuteronomy to Hittite covenants of the period 1200-1500 BC. There is a counter-argument that this could be coincidental, but some recent scholars find the resemblances convincing, and again there is no motive for forgery.
Another internal example, which I summarized here on skeptics.SE, is the word choices and writing style of the story of Exodus. Although this story contains some elements that people might reasonably find dubious (such as miraculous plagues and sea splitting), and there is no archaeological evidence of a massive flight from Egypt, the writing closely matches a historical context of 13th-14th century BC.
On the other hand, as I previously discussed here, as best we understand it today, what was practiced in Israel in the 13th-14th century B.C. probably did not resemble the strict monotheism of today's Rabbinical Judaism and Christianity. We have no certain textual evidence one way or the other, but it's reasonable to believe that the text of the Bible may have been redacted after its initial composition.
The 19th century Documentary Hypothesis proposed that the Bible was written around 800 BC. Although this held sway in the secular academy for many decades, it is no longer considered as valuable as a method of historical dating for the Bible narrative. Instead, it is used as a way to analyze the literary composition of the Bible, from a historically agnostic perspective. There are still some Biblical scholars who support historical datings of 800 BC or even ongoing redaction as late as 120 BC, but R.N. Whybray's The Making of the Pentateuch (1987) remains a strong counterpoint to overly historicist theses.