Some factors are way more important than others for rocket navigation. For instance, the azimuth, the angle you're headed is very important, since a small error in angle gets you way off target. The Saturn V theodolite, used to calibrate this angle before launch had an accuracy of better than +/- 2 seconds of arc, i.e. 1 part in 648000. That suggests you'd need to use pi to at least 6 decimal places (probably more) for surveying and azimuth calibration.

The book "Inventing Accuracy" is about missile guidance (which has mostly the same issues) and discusses the various sources of error in detail and the various contributions to the "error budget".