This crossed my mind today

In your math fetish world, it would be, but distance resolution is ultimately finite in the real world

Run a string around the perimeter lol

Depends. Can we always find a smaller resolution? Can you prove we've discovered the smallest resolution?

We cannot talk about meaningful measurements beyond the Planck length, so yes.

when you reach distances that small, its more interesting to start defining what counts as the coastline

Would the equation (perimeter/4)^2 work?

The coastline problem is cool as fuck.

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>cannot talk about
Cannot disprove their existence?

Wiki "Planimeter". If a mechanical device can do it then it can be simulated with a computer program. Then just feed in a list of coordinates for the perimeter and get the area. The more the points, the better the accuracy.

en.wikipedia.org/wiki/Planimeter