The real reason pi was never calculated to a high precision in ancient times

Even nowadays, there is no practical use for a million digits of pi, simply because we cannot measure anything in the physical world to that accuracy. For instance, one of the most precise measurements possible today relies on the Mössbauer Effect and will yield results accurate to 14 digits. That's an error of one second in 3 million years. Also note that 39 digits of pi suffice to calculate the circumference of the known universe from its radius to within the diameter of the hydrogen atom¹.

¹: Fractals, Chaos, Power Laws, Minutes from an Infinite Paradise, Manfred Schroeder, New York, W.H. Freeman and Company, 1991

Thanks to Professor Pi for explaining that "high precision" actually means only as many digits as can be used. However, the "ancients" calculated far more than they had use for. I've heard stories told about ninety-one-sided polygons inscribed in circles for measuring purposes. Let me say that again: ninety-one sides!!! That is not a sign of being too lazy to write things down.

Now let's take a look at how much accuracy was actually calculated. By using purely arithmetic methods to calculate the continued fraction expansion, pi =

             1
3 + ----------------------
               1
    7 + ------------------
                  1
        15 + -------------
                    1
             1 + ---------
                 292 + ...

Convergent   1      2         3         4           5
Fraction     3     22/7    333/106   355/113  103993/33102
Error      .045   -.0004   .000026   -8.5E-8    1.8E-10

It is clear that an incredibly adequate amount of accuracy becomes available with very little calculation. So, when the Greeks used their calculation of 22/7 as "slightly larger than pi," they had all the accuracy they needed for calculations of the time.