Even a moment's reflection reveals a problem with the
standard approach: a smaller
measuring scale is sensitive to more details. 
This is a real issue because many geological
features exhibit similar structure at finer detail.
That is, they are scale invariant,
at least for some range of scales. 
This is reasonable because the forces that sculpt
coastlines operate in approximately the same way over a wide range of scales. 
Nevertheless, surveyors measure the length of a coastline by 
selecting a measuring scale d, 
approximating the coastline by N line segments of length d, and 
deducing the length of the coastline is L(d) = N⋅d. 

Here is a picture from NASA's website. Click the picture to
see two polygonal approximations of the coastline. 

If this picture does not convince you, click here for
another NASA photograph. 
Imagine the difficulty of measuring the length of
this coastline, using smaller and smaller scales. 
Evidently, a smaller measuring scale will detect more detail of the coastline, hence
give a greater length. 
Selfsimilarity of coastlines casts doubt on the hope that these
measurements will converge as smaller scales are used. In fact, these doubts are
justified. 