If one has two parents and four grandparents, then when one goes back ten generations one has potentially 2^10th, or 1024 ancestors of the 10th generation. This becomes 1 million potential ancestors at the 20th generation and 1 billion at the 30th. If each generation is roughly 25 years, then 30 of them is 7.5 centuries, which would be in the area of 1253 AD. It is doubtful, although possible, that there were 1 billion humans on earth in 1253 - therefore everyone alive today is potentially descended from everyone alive at the time that produced descendants into the current generation.
While 25 year generations is common now, it has been traditional to marry girls at age 14, therefore most generations may average 20 years, and in that case we go back 600 years to 1413 AD. This is within a century of Columbus arriving in the Americas. By the end of this century, DNA from even the remotest settlements should be thoroughly mixed into the global population.
It would be safe to say that modern Greeks have some DNA from Greek ancestors 2300 years ago, but they will also have DNA from Hutus, Mongols, and even Incas.