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Benoit Mendelbrot described the coastline as fractal in 1967, it is interesting to see how many people have picked up on this in 43 years...
- Wikipedia quotes an exact length (11072.76 miles), then somewhat as an add on states that it is also fractal, but without any citation and no fractal dimension given. (see http://www.ben-daglish.net/mapping.shtml (http://www.ben-daglish.net/mapping.shtml) for why they chose the quite arbitrary figure).
- Wikianswers gives 2400 miles, Encyclopaedia britannica 1960 gives 2336 miles
- The British cartographic society correctly describe the coastline as fractal, but still give the figure of 11072 miles and fails to give a fractal dimension.
- Another website gives the value 6000 miles.
- The CIA factbook has it as 7723 miles.

So the idea of fractal dimensions really isn't well appreciated, despite it being essential in this question. The above values are all correct and all incorrect depending on your resolution. 11072 miles is too short if you are using a pedometer. Or if you measured it by hand with a measuring tape it might be 15000 miles long.

I think the problem is that Mandelbrot correctly showed that you can model the coastline as having a fractal dimension of about 1.2. But we have all forgotten to do the final step... 1.2 is not the amount of coastline, it is the dimension of the amount. So just as an area has units m^2, Britain's coastline has units m^1.2... but how many m^1.2 is it???

Let's say it is 20,000,000m^1.2, that means that at a resolution of 1 metre it passes through 20 million squares, and at 1/n metres it passes through 20,000,000*n^1.2 squares.

This should be how we measure coastlines, e.g. South Africa's is 4,700km^1.02. Note we are not measuring their 'length' but the 'amount' or 'size' of coastline.

This is a better way to define the size of coastlines, as people can use it appropriately, e.g. if you are planning on circumnavigating a country on 4x4 you can interpret the amount differently than if you just want to know how much string to use to mark out the coast with on a map.

We should all be using this convention right? and we could start with simple fractals. For example, Wikipedia states that the fractal dimension of a Koch snowflake is 1.2619, but what is the size of the snowflake border of diameter 1m? 3m^1.2619? 100m^1.2619?
Is there 'more' Koch curve than 2d cantor dust (which has the same fractal dimension) of the same diameter? We can know the answer to this.

I definitely like this notion. This does away with irritating problems like the fact that the length of the Koch curve is infinite--I always thought "yes, okay, but HOW BIG is it?" Same with the Sierpinski gasket, zero area when considered as a shape, infinite length when considered as a curve. The trick is figuring out a convenient way of measuring it...

Incidentally, I'm fairly sure that the Koch curve generated from a line of length 1m has a "size" of 1 m^1.2619. I like this definition a lot since by the same line of reasoning, a Koch curve generated from a 2m line has size 2 m^1.2619 :D

Also, there are lots of definitions of "fractal dimension"--similarity (aka Hausdorff) dimension, box counting dimension, point correlation dimension, etc. For simple fractals like the Koch curve these line up very nicely, but for other things (anything with irregular self-similarity, really) they tend to differ. How does using a different definition of fractal dimension change the size? I plan to look into it, but if anyone can do it faster please do :D

So now we have an extension to the whole "area of the Mandelbrot" question--what is the size of the Mandelbrot's border? The Mandelbrot's a special case since its border has Hausdorff dimension 2, so the "size" of the border is really the "area" of the border.

I totally agree. I also think that 2d Cantor dust of diameter 1 unit has a size of 1unit^1.2619, and since a Koch snowflake is made of 3 Koch curves we can say-
  There is three times more Koch snowflake than 2d Cantor dust of the same diameter.
We have a result!
With the choice of dimension type, I imagine that one could be most popular and so that is what is assumed by default, unless the measurement says otherwise. Don't know which makes the best default.

This method could be used to measure hundreds of things we don't have a simple measurement for at the moment, for example the total size of a river system in m^1.6, or the area of mountainside exposed to the air in m^2.4, or the amount of paths/roads in the US in m^1.2, or the area of coral that is exposed to the sea, etc etc.

As for what is the area of the Mandelbrot border, it would be interesting to know, it surprises me that I haven't seen any attempt to measure this, are we missing something?

Just as you can think of area as how many square blocks (or even disks) we contort to fit inside an object, you can think of the border size as how many of a standard fractal (e.g. koch curve) you can contort and line up around the border...

So after 40 years we can actually answer the title question :) (though only for west coast as that's all I could find data for):

Excellent! Though I wouldn't want to be the teacher who has to answer the question "Mr./Ms. ____, what's m^1.25?"

Something I notice is that the statement of size conveys both the size itself and the fractal dimension, when you're using the most accurate value. Mathematically speaking, if you measure with the WRONG dimension it'll go to infinity or zero as you get more precise, so you'd always need to be close to the right fractal dimension. Since the various types of fractal dimension are always close, I don't think that would ever present a problem.

Well, I bet that the issue with the Mandelbrot's border isn't that we're missing something but rather that it's so strange to ask about the "area" of a border of a 2-dimensional object. I haven't seen this application of fractal dimension before, so it's possible that the reason nobody's tried it is because nobody's thought of it yet.

And to add to the list of things we can measure: weather. Weather shows distinctly fractal patterns. We can measure the size of a cloud's surface in m^2.5 (or something), or the boundary of a hurricane in m^1.2, or the path of a weather pattern in... um... probably m^1.1 or similar.

I generally like the way this goes , but would like it to clarify the different notions of dimension it contains. Normally we do not distinguish this unless we are using dimensional analysis and even there some confuse the two sorts. This is why i generally am suspicious of dimensional descriptions. In particular the scales we construct are all fractal in any case, and we do have a choice as to what base unit we use. So for a koch curve i would take say the 5th iteration as the base and then construct a scale using self similar sections of greater and greater iterations. On the face of it this should occur every 5 iterations. Thus the finer koch scales will represent greater iterations of the basic method, and allows for a similar mathematical (scalar) treatment at first thought.

A thought experiment" take a euclidian  line and scrunch it up into a point! Or alternatively twist it around a point!

You can also count objects that traditionally either are uncountable or have quite meaningless figures.

For example, how many planets in the solar system? The main problem is that there is no obvious cut off, smaller bodies just become more and more frequent.
But using a fractal measure you can say that the number of planets is 8rM^0.5 (rM is the radius of Mercury). Ie 8 planets bigger than or equal to Mercury.
How many moons of Saturn? probably about 60km^0.35 moons.
How many rocks on a scree slope? 0.5m^0.4 ?
How many creatures on earth? 100billion kg^0.4 ?
How many cities/towns in the world? How many words in the English language? How many craters on the moon?

These are all impossible to answer satisfactorally in units^0 (which means to answer as just a unitless number) since they actually follow a fractal distribution so their natural units are not ^0.
So fractal dimensions give a way to measure anything that follows a power law (http://en.wikipedia.org/wiki/Power_law (http://en.wikipedia.org/wiki/Power_law)).

And, since practically everything seems to follow a power law of one sort or another, this gives us a lot of flexibility of measurement. I like it!

The answer to the question is 1, the units being "British coastlines" :)