For any system we could use to define a real number, (and not just because we are mortal), the system could only define or uniquely specify countably many real numbers. Any countable set of real numbers has measure 0, and therefore, almost all real numbers cannot be defined/specified using that system.

This holds for whatever system you use which defines a (possibly partial) function from [ the set of finite (but unboundedly long) sequences of characters over a finite alphabet], to real numbers

Or equivalently, from the natural numbers to the real numbers.

Not in darts. You can prove this to your satisfaction, by attempting to reach the continuum between the integers.

The dart will bounce off the metal framey bit and stick into the forehead of local barfly, 'Very Angry Ron'. Then when you graph the result, violent discontinuities will appear in the function, across a wide range of scales.

These already have their own special set in category theory and are sometimes referred to, within the field, as; 'Assorted and Unreasonable Associates of Very Angry Ron'. That is, the other side of a reasonably big field, ideally with some cattle in the way.