Separate from the fact that the Bible does not in any straightforward way try to "date" the moment of Creation (that is an extrapolated calculation one can make from assumptions relating to the "X begot Y" sections and whatnot), it is just worth considering that this length of time (10,000 years or so) coincides fairly well with what we know about when the transition from Paleolithic to the Neolithic period happened in that part of the world. Which is to say, it corresponds very well with the shift in human activity and culture that took place as people moved from nomadic, pastoral societies and started in on more long-term habitation and agriculture. This has led some speculative writers to read Genesis as not an account of the "beginning of the world" but as a filtered version of accounts of how people became less "tribe-like" and more "society-like," that the fall from grace is really about people changing their relationship with nature, etc. But this gets well beyond my ken.

Note that stuff older than 10,000 years or so is pre-literacy, pre-writing. If you do not have writing then you are reliant on things like oral traditions to keep historical records. These are notoriously less precise, even with forms (e.g. rhyming schemes, epic poetry, other mnemonics, etc.) that are meant to encourage fidelity in passing on such stories. The first written records only date from around 6,000 years ago.

To put it into further perspective: most people alive today, despite having immense ability to keep track of records and genealogies, cannot name many family members older than their grandparents. How many people know the names of their great-grandparents, much less their great-great-grandparents, much less great-great-greatgrandparents? My grandmother was born in the 1930s; her mother was probably born in the early 1910s; her grandmother was born in the 19th century. It is nontrivial to trace generations back a whole lot further than that — and it's pretty damned hard to imagine doing it without written records. We have some records of families and genealogies that go back thousands of years (think about the Chinese dynasties) but it is non-trivial and the sort of thing that is generally done on the level of states, most of a much later period than what you are talking about.

A more interesting question to ask is, why did so many smart people after the writing of the Bible believe that date for such a long time? I mean, OK, we can say that the people who wrote the Bible might not have had great archeology. But why did, say, Isaac Newton and plenty of other very intelligent, scientifically-minded people believe that date? It wasn't just "blind faith" — there really isn't a lot of "easy" evidence for human habitation earlier than, say, 6,000 BCE. The oldest "easily visible" structures, like the pyramids of the Old Kingdom of Egypt, date from around 2,500 BCE, and they were considered incredibly old to people like Aristotle (e.g. 340 BCE), much less people living in the 17th century (much less today). Again, there are really good reasons for this lack of "easy" evidence — not much sticks around that long unless it is large, durable, and to make large and durable things requires quite a lot of social organization (which is why monumental architecture is characteristic of the Urban Revolution, where you start to get kings and states and whatnot). There are a few contrary examples of this (Stonehenge, for example, is now known to be from about 3,100 BCE, so older than the pyramids) but it is interesting to note that dating any examples developed prior to writing is pretty hard (there were those in the 17th century who thought that Stonehenge was some kind of Roman ruin, for example — it doesn't exactly explain itself or declare its age).

The fact that the Earth is much, much older — and that human habitation is much, much older — than the time of the Urban Revolution (much less Neolithic and Paleolithic) takes a lot of observational work (e.g. archeology, geology) but it also requires a lot of conceptual work (e.g., taking into account the impossibly long amount pre-human history). It was not until the 19th century that these ideas really caught on, and this work really bore fruit, and people started looking for the immensely old. Martin Rudwick's excellent Bursting the Limits of Time (2007) is all about this revolution in human thinking, this difficult realization that the world is much, much older than we are, and that even we are older than we have given evidence for. Even then, many of these people weren't anti-Biblical, they were just anti-literalist (Louis Agassiz, for example, is someone who essentially believed in special creation, just not in the literal account of it in Genesis).

Always more to be said on a topic, but this old response by /u/algernon_asimov would likely be of interest to you.