Would it have been possible for the ancients to have built a Tower of Babel? As recorded in Genesis 11:1-9, the ancients allegedly built a tower to the skies in a valley in Babylon (near present-day Baghdad), and God scattered them across the Earth as a punishment. As discussed over on Judaism.SE, the tower was either ~52.5 km tall or ~2.6 km tall. Are either of those numbers remotely possible, given the materials and technology of the time and place? Note that Orthodox Judaism places this incident 3781 years ago, in 1765 BC.

Given that it is possible that some building techniques were lost to history, we'll never been able to definitively answer this question, but we can take a look at what we are capable of building now to see how it stacks up.

Taking a look at the wikipedia entry for tallest man made structures we can see that the tallest structure ever documented is the Burj Khalifa, standing 828.1 meters, in Dubai. Using your lower estimation of 2.6km we can see that the Tower of Babel would have been more than three times as tall as humanities current tallest building.

I'm not an engineer, but I'm very comfortable in saying that in 1765 BC humanity lacked the ability to build a 2.6 km building