I know how to use a ruler to approximate a length of an object (like a wire or a stick) in meters. I could also use the ruler to approximate a two dimensional area (like a table top or a parking lot) in $\text{meters}^2$ by dividing it into a grid and counting squares.
I read that we can estimate the length of the coast of Britain to have fractal dimension 1.25. Is there a value in $\text{meters}^{1.25}$ giving the 1.25-dimensional fractal measure of that coast? Can I calculate it using my ruler or rulers of different sizes/precisions? Or if I drew a Koch snowflake whose largest triangle had side length $1\text{ meter}$, could I find it's $\ln(4)/\ln(3)$-dimensional measure in some analogous way?
If there is such a thing as fractional-dimensional measure for dimension $d$, can we give a fractional unit like $\text{meters}^d$ physical meaning?