See these 2 paradoxes:
Coastline paradox
The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length.
http://en.wikipedia.org/wiki/Coastline_paradox
Dichotomy paradox
Suppose Homer wants to catch a stationary bus. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a quarter, he must travel one-eighth; before an eighth, one-sixteenth; and so on.
http://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Dichotomy_paradox
Question
These paradoxes state that we can't measure these distance because as we look at smaller and smaller scales the distance increase to infinity. Are these paradoxes limited by the size of an atom?
Let's suppose that we can see the atoms of the coast:
O
OO
OOO
OOOO
OO
Land OO
OO Sea
OOOOO
OO
O
Then we can measure the length at the right of the coast, for example by tracing a curve which join each consecutive atom:
\
O\
OO\
OOO\
O\
Land O|
O/ Sea
OOOO/
O/
/
This would make a finite length in theory (even if it would be very hard to measure in practice). Is it relevant to measure the distance under the size of an atom?
If the size of an atom is relevant, then is the Plank Length the limit?
according to string theory lengths smaller than this do not make any physical sense
The coastline paradox is a way to illustrate fractals. So it's not intended for the physics world. If you go into the physics world, all the particles in the "perimeter" of a coastline are constantly moving, so an exact answer cannot be given but an upper bound can be found using Plank's length.
The Zeno paradox lies on the assumption that an infinite amount of steps require an infinite amount of time. But that's obviously false. If Homer is walking at a constant speed, then he'll walk half the distance to the bus in $t$ seconds. Then he'll walk half the remaining distance in $t/2$ seconds... at the end, he'll take $\sum_{k=0}^\infty \frac{t}{2^k} = 2t$ seconds to get to the bus.
Answering the question in the comment as to what I mean by constanly moving particles I wrote this:
As far as I know all particles have momentum. The warmer they are, the faster they go, and even if you could get a particle at absolute zero (wich can't be done) it has what is called 0-point energy that gives it still some motion. And if you "freeze" the coastline so the particles stop moving, then you would know it's momentum (it would be 0 since they don't move) so by Heisenberg's uncertainty principle, you wouldn't know where the particles are, so you couldn't make any measurement.