How do Young Earth Creationists account for the fact that some cultures have been continuously measuring the years for almost five millenia?
Lining up dates from ancient documents and inscriptions is not an easy thing to do. You don't find ancient records that say, "Our nation was established on June 12, 3462 BC", for the simple reason that our modern calendar did not exist back then. So even when an ancient source gives a date, it is in the calendar they used at the time, and historians have to try to match that up to modern calendars. Scholars debate these sort of things endlessly.
So the reality is not that ancient Egyptian or Hittite or Sumerian or whatever chronologies contradict the Bible chronology. Rather, it is that SOME INTERPRETATIONS of ancient Egyptian, etc chronologies contradict SOME INTERPRETATIONS of the Bible chronology. Put that way, it's hardly an astounding fact. Reporters trying to describe current events often have difficulty fitting together all the different accounts of an event into a single coherent narrative.
When people talk about taking Bible dates literally, they usually eventually refer to James Ussher, who used the genealogies and other information in the Bible to estimate the date of creation at about 4004 BC. Modern young-earth creation theorists generally accept that as at least a ballpark, though many would stretch it out a few thousand years. Some secular historians claim to trace historical records back to circa 6000 BC. So yes, there's a discrepancy to be resolved, but it's on the same order of magnitude. No one claims to be able to trace historical records back a million years or a billion years. The histories of cultures around the world are at least generally consistent with young-earth Biblical chronology. There's no way you could use them as evidence for an evolutionary time-scale.
If these "historical records" could be trusted as absolute truth then this would be a problem for Young Earth Creationists.
However, ask any ancient historian and they will tell you that there has been some major corruption in the so-called "historical records" of ancient cultures. This is due in part to the "age wars" which took place between various ancient societies, each of which was attempting to show their people were the first people!
Remember - if you dig up an ancient scroll or tablet you don't know the words on it are true. Historians have to make assumptions just like scientists, theologians, philosophers, and so on.
I have heard secular historians state that Jewish history is generally considered to be the most reliable. (Of course, they wouldn't include the "creation week" or "worldwide flood" due to the modern scientific position that these things didn't happen. But that's another topic.)
The Young Earth Creationist considers the Bible to be the only historical record which is absolutely true and reliable. Therefore, the YEC (by the same logic) could ask "how do skeptics account for the fact that their beliefs about history conflict with the objective truth of Scripture?" But that would be rhetorical. :)
Clarence Larkin's chart on the table of nations suggests that Noah himself, after the flood, was the father of the Asian races. His lifespan happens to date back 5 millenia. So, it certainly would be possible that the Chinese date back five millenia.
The fact that their language changed at the Tower of Babel does not mean that all history was lost at that time either. The people at that time didn't just forget everything that happened before their language was confused. They very well could have all been keeping track of dates since Noah, the father of all the earth, was born.
Admittedly, we can't know for sure at this point, but this is certainly a reasonable possibility.
While is true that there are cultures that have written dates prior to the Tower or the Flood, it is also true that there are cultures such as the Mayans that date their calendar in "days since the flood" and which are quite close to the dates from Scripture.
The Mayan stories of the Popol Vuh, while not containing an ark story, do contain a universal flood and I find it quite interesting that their dates are so similar to biblical chronology.
My point is, this issue is something of a non sequitur. Certainly there are records with wide variance from Scripture, but there are also records with quite similar timelines.
Biblical literalists and other Young Earth Creationists most often account for these facts by seeking to cast doubt on the careful research of historians and archaeologists. They may claim that the Egyptian, Hittite or Chinese cultures are not as old as the evidence seems to show.
Written records exist in the context of archaeological records and one ought not examine either one without the other. Ofer Bar-Yosef says in 'The Natufian Culture in the Levant, Threshold to the Origins of Agriculture' late hunter-gatherers inhabited the Near East until about 13,000 B.P. (Before Present), when sedentary Natufian hamlets began to emerge in the Levant. Wikipedia reports the earliest settlements in Eridu, a conglomeration of early temple-cities in Sumer, southern Mesopotamia, dating to around 5000 BCE. Clearly these are much too early for the biblical Flood, which supposedly occurred 2348 BCE, or about 4363 B.P. Biblical literalists generally ignore these early settlements and concentrate on later, more well known civilisations such as in Egypt or China, where it is more plausible to suggest that they post-date the biblical Flood. Nevertheless, the start of the Egyptian Old Kingdom is dated to 2649 BCE, hundreds of years before the Flood. The usual response is to cast doubts on the archaeological evidence rather than accept a redating of the biblical narrative.
No one claims to be able to trace historical records back as far as the archaeological evidence, simply because historical records are written records, and writing was not developed until around 3300 BCE, with the introduction of cuneiform script in Mesopotamia. Writing began to provide a continuous record of events from before the time attributed to the biblical Flood. Some concede there's a discrepancy to be resolved, but it's on the same order of magnitude because no one claims to be able to trace historical records back a million years. That is the role of archaeology and anthropology.