We know that $0 = \lambda f. \lambda x. x$, $1 = \lambda f. \lambda x. f(x)$ and so on, with signed numbers representable as a natural number and $0$, and rational numbers can be represented as pairs of integers.
So how about irrational numbers like $\sqrt{2}$ and $\pi$? Can they be represented exactly in lambda calculus? Like, for example, $\sqrt{2} = \lambda f. ???$ (I don't want to take $\sqrt{2} = 1.414213562$ (the Wikipedia page says something the same or similar) or something like that, please just give me $\sqrt{2}$ as it is).
There are only countably many λ-terms (see here or there), provided you consider a countable set of variables. Hence there can't be an injective encoding of real numbers in the (finite) λ-calculus.
You could consider an infinitary λ-calculus (see the original paper, or Chapter 6 of the Satellite). The set of Böhm trees, or the 001-infinitary λ-calculus are enough. In this case, there is a straightforward encoding of $[0,1]$: $$0,a_1a_2a_3... \mapsto \underline{a_1}( \underline{a_2}( \underline{a_3}(\dots)))$$ where $\underline n$ is an encoding of the digit $n$ (whichever kind of “digits” you like: you could just take $\underline 0=λxλy.y$ and $\underline 1=λxλy.x$). Then you can tweak it to get an encoding of $\mathbb R$.