It would seem we don’t know exactly what method they used.
I would like to propose: really, really long and accurate ruler, just because the mental image is funny. The Greeks hadn’t come around yet to make everybody’s homework harder by insisting on compass and straight-edge (Note: this comment is not historically accurate).
I would agree on the long and accurate ruler. The actual problem here is understanding what the square root is (and how it can be used), not finding it's value for some arbitrary number. And despite all even 4000 years ago our ancestors could make a very precise tools.
Entirely agree. To be honest, I think it rather marvellous that people were able to imagine, then reason about, so much precision so long before any physical representation of it was possible. The Babylonians could calculate to a ten thousandth of an inch, but no human could measure the physical equivalent until Henry Maudslay. In a way the whole Industrial Revolution rested on people realising that the five-thousand-year-old mathematical view of abstract truth of "shape" could be rendered as physical reality with the right techniques... but coming up with those abstractions in a world where the roundest thing came off a potter's wheel and the basic unit of length was some fellow's forearm was an even more impressive achievement. We should not take it for granted just because it was so long ago.
Why? Long rope, make a right triangle. The proportion of one of the equal sides to the remaining side is square root of 2. Seems more likely to be experimentally approximated before calculated.
This would be useful for figuring out land rights. And for that, you need very long ropes.
I would be shocked if anyone could measure sqrt(2) to half a part per million using ropes.
You'd need to be accurate to a half mm per km of rope, which is well beyond what you need for practical surveying. You also need to keep the rope essentially perfectly straight, which means doing the experiment on almost perfectly flat ground. You'd need to keep the rope at a controlled tension so it doesn't stretch. You'd be hosed by temperature and humidity changes while walking those kilometers. The rope would also need to be marked or measured to < 1mm accuracy when you're done walking. You'd need to make sure the right angle is square to < 1ppm as well, though maybe there's a clever procedure that corrects this.
It's not even easy to do this with laser surveying equipment.
Real-world ropes flex and bend, which alone causes experimental error orders of magnitude larger than one in a million. There does not exist a rope material that doesn't flex at least one millimeter per a km of rope, and certainly did not exist 6000 years ago. Hell, even if we assume a perfectly flat plain, the curvature of Earth would cause an error larger than 1e-6. And obviously if you try to hang the rope from both ends and pull it taut, it will flex and it's also impossible to remove all the slack.
You need a laser, or at least something like the Michelson–Morley experiment, exploiting the wave nature of light, to measure any real-world distance to the precision required. And obviously for that to work, you also need to know the speed of light to the same precision. And given that we're not in a vacuum, even light bends as it travels through air, especially air near the ground. You know heat haze? Yeah, that's going to mess up your experiment too.
And how on Earth would they ensure the ropes form a right angle to a precision of one in a million? Given a 1km long side, the sideways margin of error is again 1mm. The angular precision needed is sub-arcsecond, a resolution that you need a large-ish optical telescope to achieve.
And obviously then there's the question of measuring the length of the hypothenuse.
