We see a lot of papers and talk about ancient Babylonians exactness of calculating the value of square root of 2. For example: http://johncarlosbaez.wordpress.com/2011/12/02/babylon-and-the-square-root-of-2/
But how close could they approximate the value of SQRT3? I have a document that talks about the subject, but I cannot trace the answer directly from it: http://www.helsinki.fi/~whiting/roots.pdf
Maybe someone has knowledge of the method they used and reference to the old Babylonian clay tablets, where it can be verified and seen in use.
The tablet MS 3051 (Friberg, A Remarkable Collection Babylonian Mathematical Texts) deals with the calculation of the area of an equilateral triangle; the answer given there leads to the simple approximation $\sqrt{3} \approx 7/4$. This does not mean that they couldn't have done better.