What is randomness? It is very to give problems of the sort: I have a set $S$ and all elements of $S$ is a certain value assigned. For example it is very easy to ask, how likely is it, that a thrown common, fair dice shows the number {6} . 
Things become a bit more tricky, if the set becomes infinite, but even then somehow one can deal with it. I view it like this: One has the value of 1 and somehow has to distribute it all over the elements of this set. 
Though often it is the case, that one has to differ between the variables that have a certain property and those that don't. 
For example: How likely is it that a thrown, common, fair coin never shows tail? This questions is the same as asking: How likely is it, that you get head at your first throw, at your 2th throw, at you third throw and so on.
But what is someone doing, if this distribution can not be so simple obtained? For example: There is a target in form of the unit circle, one throws a dart at the target and now asks, how likely is it, that it hits a point, with rational coordinates? Since $\mathbb{Q}$ is countable and $\mathbb{R \setminus Q}$ uncountable, I guess, the likelihood hitting a point with rational coordinates equal $0 $.. 
I am guessing, because I don't know. How does one define such probabilities? Is there even a good definition to begin with? 
 A: What you are probably looking for is continuous probability distributions
which deal with this type of phenomena. The usual approach is to just pick some reasonable measure on the space in question (the unit disk in your example) and work with that. The downside is that the probability of "hitting" any given point will usually be zero. The upside is that you get reasonable behavior if you start working with sets of points and given the fact that in the real world you never "hit a point" you always hit some area if only because the size of the tip of the dart is non zero, this works just fine for most applications.
It's worth noting that the transition from "real world" probability to Probability in mathematics is quite complicated from a philosophical point of view.
Edit:
In response to some questions in the comments I will try and go a bit further into the idea of both continuous distributions and mathematical modeling of some real world "random" occurrences. 
The first thing to think about when you are considering any mathematics is to know what tools are at your disposal. Let's quickly recap finite, discrete and continuous distributions and probability spaces.
In general the definition of a probability space is the following:
Let $\Omega$ be a set, let $\mathcal{F}\subset \mathcal{P}(\Omega)$ be a $\sigma$-algebra. That is:


*

*$\Omega\in\mathcal{F}$

*if $A\in\mathcal{F}$ then $X\setminus A\in\mathcal{F}$

*$\{A_i;i\in\mathbb{N}\wedge A_i\in\mathcal{F}\}$ then $\bigcup_{i\in\mathbb{N}}A_i \in\mathcal{F}$


(In words that means $\mathcal{F}$ is closed under complements, countable unions and contains the whole space.)
Let further $P:\mathcal{F} \rightarrow [0,1]$ which is countably additive and $P(\Omega)=1$. In math terms:


*

*$S=\{A_i;i\in\mathbb{N}\wedge i\neq j\implies A_i\cap A_j=\emptyset\}\subseteq\mathcal{F}$ then $P(\bigcup S)=\sum_{i=0}^\infty P(A_i)$


We call the triple $(\Omega,\mathcal{F},P)$ a probability space.
Notice that this is an extremely general definition. You can fit nice finite probability spaces, like the one you get by throwing a die, and also quite complex probability spaces like the space of times you wait for a bus on a bus stop (which is continuous).
When considering some real life problem, the first issue is deciding how to model it mathematically. In the case of throwing a six sided die the method of modeling is quite straightforward. You consider $\Omega=\{1,2,3,4,5,6\}$ the possible outcomes, $\mathcal{F}=\mathcal{P}(\Omega)$ the $\sigma$-algebra which just turns out to be the whole power set and $P$ is defined on each single element subset as $P(\{i\})=1/6$ and extended to a complete function on the $\sigma$-algebra in the obvious way.
If you have the case of throwing darts the issue becomes a bit more muddled. The obvious base space would be a subset of $\mathbb{R}^2$, but beyond that it gets quite muddled. This base space is not only infinite but also uncountable which complicates the way we can define a countably additive measure. 
The first thing we should consider doing is choosing a reasonable $\sigma$-algebra. The standard case would the borel $\sigma$-algebra which captures pretty much everything we might be interested in modelling from the real world. 
The next issue is how to define the probability function. If we want a function which is "translation invariant" (i.e. the probability of hitting a region does not depend on where it is but only how big it is) then it's easily seen no singleton from the base space can have positive probability. This actually looks worse then it is, since if we are modeling the real world then certainly we can never "hit" a single point. We always hit some area. This is not really because the tip has nonzero size but rather because out ability to measure precisely is limited. Whenever you properly measure something in physics you really get a number along with some error bars. 
Thus being able to say what the probability of hitting an area should be is a decent enough probability function. Assuming we have a uniformly random thrower then the probability function which will best model the process is just size of the event area over size of full area. 
Another approach to the last problem is to instead of an area consider a grid and then you end up with a finite state space.
The last thing I want to write a little about is the problem that cr001 poses in the comments. What happens if you ask two people to choose a number at random from the interval $[0,1]$. If these are ideal mathematical people then as cr001 points out the probability that they both choose the exact same number should be $0$ assuming they are choosing with respect to a uniform random distribution on $[0,1]$ ($P([a,b])=b-a$).
This seems wrong because intuitively we feel that if we actually ask two strangers to pick such numbers there is a very much non-zero chance their answers will match.
The reason for this intuitive mismatch is easy to explain. Random stranger do not choose even remotely uniformly at random from the interval $[0,1]$. In the first place this can not be humanely done since most of the elements of this interval are not even recursively enumerable, and thus certainly can not be described. So such strangers would at best be able to choose from some countable subset of the $[0,1]$ interval. But reality is much harsher while I haven't done any research I would bet a decent amount of money that over 90% of people will choose a number which can be written as either a fraction with numerator and denominator of less then 4 digits each or a similar multiple of $\pi$ or $e$. Thus the actual probability function should be 0 almost everywhere with non-zero probabilities on some finite set. Moreover I'm sure an easy experiment would show that the probability is strongly non-uniform even on the non-zero probability set with things like $\pi/4$ or $0.5$ being much more common then say $723/418$.
