It is possible to define our intuitive notion for probability in subsets of $[0,1]$ I've always heard and read the sentence:

If you pick a real number $x\in[0,1]$ at random, the probability to obtain a rational number is $0$.

What is the meaning for that? Is this the "real" probability?
First, we need to define accurately what means at random, I'll assume a base $10$. Then all numbers in $[0,1]$ begins with $0$ (for $1$ we discard this notation and we will use $0.9999\ldots$).
Then I'll give a classical approach defining random meaning, if we pick a real number $x$ in $[0,1]$ at random we mean that the probability to get a digit in certain position is $1/10$ (extending for $[0,1]$ the Laplace classical definition for probability), roughly speaking each digit has the same "probability" to be in $x$.
Strictly speaking we are building a notion of probability in $\{0,1,2,3,4,5,6,7,8,9\}^\omega$ since numbers with a $0$ periodic has two representations. We want to pass the intuitive notion of probability to this set, but if there is one, this must be unique.
The Lebesgue measure $m$ rescues some of these probabilistic notions for example $m(\emptyset)=0$, $m(\{x\})=0$, and $m(\text{numbers starting with the given digit $k$})=\frac{1}{10}$, but intuitively a probability notion must be defined in all subsets (Edit: sos440 in comments says that is natural this phenomenon (not defined in all subsets) since we can't describe some sets without using additional axioms like "Zorn Lemma") apart of having some properties like (finite or $\sigma$)-additivity.

Lebesgue measure also satisfies $m(\mathbb{Q\cap [0,1]})=0$. Does our intuitive notion of probability tell us that the probability to get a rational number is $0$?

Other functions("measures") in $[0,1]$ are defined in whole $\mathcal{P}([0,1])$ but doesn't coincides with $m$ and we loss other notions for our intuitive probability meaning.

Does Lebesgue measure deserve to be called a "real" probability in $[0,1]$ (or $\{0,1,2,3,4,5,6,7,8,9\}^\omega$)? Why?
This was completely answered by Michael Greinecker ♦ The answer is a big Yes.
Is possible to define a probability notion for $[0,1]$ and for whole $\mathbb{R}$?
Fully answered in comments and answers. The answer is no for $\mathbb{R}$, and for $[0,1]$ is yes.
Is there any function(measure) that is the best representative for our concept of probability? Why?
This was completely answered by Michael Greinecker ♦: The Lebesgue measure.

Thanks in advance.

Second Edit: People in comments and answers talk me about importance of additional axioms like "Axiom of Choice" and "Continuum Hypothesis". How acceptance or denial of these axioms affects the answers to my questions here?
This was fully answered by hot_queen.

Third Edit: I marked the questions were completely answered and offered a bounty for intuitive answer and essencially the first question about $\mathbb{Q}$.
 A: First, under the continuum hypothesis, every probability measure on the powerset of $[0,1]$ is concentrated on a countable set of numbers, so it would give a highly asymmetrical notion of "at random".
Second, $\mathbb{Q}\cap[0,1]$ is countable and any measure that does not put positive mass on some numbers will put zero measure on countable sets. Every probability measure that is invariant under all permutations of countably many points will put zero probability on $\mathbb{Q}\cap[0,1]$.
Third, Lebesgue measure is in some sense the natural notion of a uniform distribution. Let $X_n$ be a random variable with uniform distribution on the set $\{0,1/n,2/n,\ldots,n/n\}$. Then the sequence $(X_n)$ converges in distribution to a random variable with Lebesgue measure as its distribution.
Ultimately, a probability measure deserves to be called random if it doesn't discriminate between events we consider equivalent. If we think of $[0,1]$ as being wrapped around a circle and think that shifting a set along the circle should not change its probability, Lebesgue measure is the only measure (on the Borel sets) satisfying this criterion. There is no similar probability measure for all of $\mathbb{R}$.
A: I wans't planning to write a "rant" but it turns out that my beer is pretty good.
Mr. Greinecker talked about a scenario (CH) where it is impossible to measure all sets of reals (this was shown by the Banach and Kuratowski a long time ago late 1920s maybe). Let me add that if you do not require translation invariance then it is conceivable that all sets of reals could be assigned a measure. One way of "justifying" this is to say that people have worked in the theory ZFC + "there is such a total measure" for quite some time now and have never obtained a contradiction. The other way of justifying this is that they "feel" that in their set theoretic universe ZFC + "a measurable cardinal exists" holds and hence by Solovay's work the former assumption must be consistent too.
Of course the moral is "it is difficult to compute the truth content of infinitary statements" (Nelson believes that PA is inconsistent) and once formalized in ANY decent system, not only these questions but their independence becomes independent (over your decent system).
This is a day-to-day phenomenon for logicians (set theorists, in particular) and maybe someday all mathematicians will ask not just for a yes or no answer to their conjectures but also an independence result (ZFC in my view exhausts my intuition of truth and knowable). 
A: If the intuitive notion of probability in the interval is that subintervals of equal length contain the sample point with equal probability, then it is intuitive that the rationals have measure 0. Take a small, small interval around 1/2, say of width $\epsilon$, then one half that big around 1/3, etc. and go on until you've surrounded every one of the countably many rational points by a collection of intervals whose total length is 2$\epsilon$. By our intuitive notion, this means that the probability of getting a rational point is less than $2\epsilon$ for any positive $\epsilon$, so it must be 0.
A more intuitive answer is this: what is the probability (pretending things can be zoomed up on infinitely far) that a wooden board is exactly 1 ft. long? What about 45/37 ft. long? Intuitively the board will always be a little off of any perfect fraction, meaning it is irrational.
