On the NASA front, I believe I read somewhere that NASA determined that only 40 decimal places of pi are required to define a sphere the size of the observable universe to the accuracy of +/- the width of one hydrogen atom. It seems like you could file that under "close enough."

Just using 3 is certainly too low of a precision -- unless you're writing a major work of religious literature, of course. 3.1 is likewise unlikely to result in acceptable accuracy on a terrestrial scale. I've always used 3.14159 which is conveniently exactly what I can remember without looking it up and it's always been good enough for me. I don't think I've ever in my life needed to scribe a circle much larger than a couple of feet across at any rate.

You may be interested in reading this: https://blogs.scientificamerican.com/observations/how-much-pi-do-you-need/

You can calculate this yourself. For example, if you're working on an object that's 1m in diameter and you use 3.14 to compute the circumference, then you can expect errors of up to 1m * (3.142 - 3.14) = 0.002m = 2mm.

Not a lot, and this is why to speedup thing on some architecture, when working with (unsigned)integer you multiply by 355 then divide by 113 (it's like 3.14159292035)

The answer to your question is as simple as it is unsatisfying. Additional degrees of precision in construction are only as useful as your means to use them.

If you using your saw can only cut to an accuracy of 1/8 of an inch, than any precision beyond that point is lost as you are unable to actualize it

However if you are using a saw and you're now at the point of your personal skill that you're measuring to the inside or outside of the mark on a ruler, then it is likely time for you to graduate between more precise form of measurement.

https://youtu.be/qE7dYhpI_bI?si=HCtTbklCA18ZieCh

This video covers a lot of the interesting points around measurement and how we can never truly be perfectly accurate with any measurement of any non-discreet metric.

To give a real world example if you are off by a millimeter diameter when building a car engine cylinder it will likely fail.

Not a direct answer to the question but one thing not noted in other answers is in computing you often work at a higher precision than you need for your final answer as the errors tend to increase each time you do a mathematical operation.

In the world of reasonably powerful hardware (laptops, desktops, servers, smart phones etc.) we'd typically work with 64 bit floating point numbers which gives pi to 15 digits (I think, not at a real computer now so can't check). because it's simple to do so even though we don't need the full precision.

It's been about 20 years since my engineering degree, but we used to memorize to six decimal places. Anything more than that is never used.

When someone wants to be more accurate, they add another digit.

So, in terms of accuracy required, an old structural book I have includes a "functions of numbers" section which has various things including the square, the square root, the log, and the circumference of a circle with that diameter. It shows pi as 3.142. Outside of alignments, I would expect that to be good enough for most civil engineers.