Euclid's Elements treat as axiomatic the proposition " the whole is greater than the part" .
Set theory is said to overthrow this so-called self evident truth, because for some sets ( namely infinite sets) the part ( proper part) has the same size ( cardinality) as the whole.
What is exactly the impact of this set-theoretic result (1) on geometry and (2) on mathematics in general?
Is it actually the " blow" it is sometimes said to represent for mathematical rationality?
These kind of problems usually arise from applying different definitions to the same situation, so I will try to make things as clear as I can.
When Euclid wrote "the whole is greater than the part", he had a point of view of mathematics very different from our point of view. All of ancient Greek mathematics, actually, was heavily focused on geometric problems; and, given the fact they had much less "mathematics" than us, their problems were, necessarily, simpler as well.
Euclid wanted mainly to, and here I am not criticizing his scope, measure and build geometric entities as precisely as possible (and the "as precisely as possible" was indeed the revolutionary part of his work). So, if you take any reasonable segment of a finite straight line, of course you see that what you obtained, the part, is less than what you started with, the whole.
Now, let us move to a slightly more modern view of mathematics, which includes, e.g., the real numbers. What is a reasonable segment of a finite straight line? Well, the closed interval $$[0,1]=\{x\in\mathbb{R} : 0\leq x\leq 1\}$$ is, arguably, a finite straight line, and $[0,1]\setminus\{1/2\}$ is, also arguably, a reasonable segment of a finite straight line. And yet, you could not distinguish them with a ruler: the case here is that, for Euclid, this would not be a reasonable segment of $[0,1]$, given, among other things, he had not a clear concept of the real numbers, so you cannot look at this problem from his point of view and expect his axiomatic to work flawlessly.
You can then say "OK, but $[0,1]$ IS larger than $[0,1]\setminus\{1/2\}$, in the sense that the former is a proper subset of the latter", and then it becomes natural to look at things from Cantor's point of view. Cantor's work, or rather its later formalization, is exceptionally precise, and statements such as "$X$ is greater or equal to $Y$" have a very specific meaning.
In set theory, this would mean (ignoring Axiom-of-Choice-related problems) that there exists a surjective function (i.e., such that, for every $y\in Y$, there exists $x\in X$ satisfying $f(x)=y$) $f:X\rightarrow Y$; two sets $X$ and $Y$ are then of "equal sizes" whenever one is greater or equal to the other and vice-versa, what comes down to there existing a function $f:X\rightarrow Y$ which is both injective (that is, if $f(x_{1})=f(x_{2})$, then $x_{1}=x_{2}$) and surjective.
Surprise, surprise, in set theory, $[0,1]$ and $[0,1]\setminus\{1/2\}$ are both of the same size! But, as explained, this would probably not be a problem for Euclid: the ends, and means, of his axiomatic did not involve the treatment of pathological things as these. Another example, that may make things clearer, is Galileo's paradox: think of the natural numbers $$\mathbb{N}=\{0, 1, 2, 3, 4, 5, 6, ...\}$$ and its subset of even numbers, $E=\{0, 2, 4, 6, ...\}$. To Euclid, to measure both of these sets had no meaning, they were not geometrical objects; to the mathematicians just before set theory, it was clear that $\mathbb{N}$ is larger than $E$, given it contains all elements of the latter, and so many more! But, in set theory, and using the definition found in set theory, both are of the same size: take $f:\mathbb{N}\rightarrow E$ such that $f(n)=2n$.
So, to answer your questions:
(1) Huge. Set theory has altered all areas of mathematics deeply, some (algebra, topology come to mind) more than other (geometry), but it did not "overthrown" it. Of course, we have indeed some problems when putting the two together (Banach-Tarski's paradox is an example, coming from confronting intuition with counter-intuitive tools such as the axiom of choice), but set theory actually helped solidify geometry as it stands today. (2) More than huge. This specific result is a tiny part of set theory's many, sometimes apparently absurd, contributions to modern math, all steaming from Cantor's attempt to measure sets.
No! As I explained, people that think this "destroys" our rationality are usually being too dramatic or applying different lines of thought to the same problem. Yes, set theory is sometimes counter-intuitive, but how really good is our intuition anyway? How many times have you seem an infinite collection of things and operated with it to gain that intuition?