When you need that kind of precision for irrational numbers, you can opt for symbolic calculation, where important irrational numbers are simply symbols or terms and you can transform them at the end of the calculation into a number of as much precision, as you need. Of course there are always edge cases, where finding a good solution is difficult. For example, what if you need performance and as much precision as possible? Cannot use floating point numbers, cannot use rationals with same performance as floating point numbers (I think), cannot use symbolic calculation, also because of performance ... Not sure what to do then, but I have not hit such a case ever. I imagine, that perhaps space operations need that kind of precision. Perhaps they simply have clusters run for a few days or so, to get to that precision.

They also can come in handy in various puzzle problems.

Outside a laboratory setting, there are very few real world problems that can't be adequately addressed with scaled integers (fixed point math).

Most things have practical limits imposed on accuracy. Most monetary transactions are legally required to be settled in pennies (at least in the US). Most time periods are recorded in minutes or seconds. Most thermometers are only accurate to 1/10 of a degree. Attempting to exceed these practical, real world limits is really just "false accuracy" that causes more problems than it solves.