We all know that the idea of irrational numbers was very much anathema to the Pythagorean school of mathematics, and that its revelation caused quite a stir. But ultimately, after the dust had settled, the Pythagoreans had to accept the proof and move on. How, then, did they incorporate this radical new idea into their mathematics? How did they account for it philosophically?
The answer to this discovery was the theory of proportions (Euclud book 10), where geometric language is used to deal with numbers (and numbers themselves are banned). The theory of proportions was a substitute for the modern theory of real numbers. It is not equivalent because not all numbers can be constructed with a compass and a ruler, but Hellenistic scientists did not know this. Eudoxus and Archimedes developed a method of exhaustion which is essentially equivalent to the modern theory of real numbers. (They understood the main property of the real numbers which we call "completeness").
It took long time to develop the modern theory starting from Nepier and until Cantor. But notice that physicists of 17 and 18 centuries formulated their laws in terms of proportions, not numbers! And this formulation survived till the middle 20-th century school textooks. "The gravitational force is proportional to the mass and inverse proportional to the square of the distance", etc.
Irrational numbers were not anathema to Pythagoreans, they never thought of them at all, or of rational numbers for that matter. The only numbers they acknowledged before, during, and after the discovery of incommensurables were positive integers. Euclid in Elements writes "ratio is a sort of relation in respect of size between two magnitudes of the same kind", i.e. ratios are not numbers, nowhere in Elements are ratios added or subtracted, in fact they never appear alone, only as sides of a proportion. He also writes "commensurable magnitudes have to one another the ratio which a number has to a number", i.e. Pythagoreans discovered the possibility that not all ratios were as integer to integer.
It is likely but by no means established that they originally thought otherwise, largely because that would simplify the theory of similar figures. However, there is no evidence that Pythagoreans were disturbed by the discovery. Even before incommensurable ratios were studied for their own sake, Hippocrates of Chios and Archytas of Tarentum used them in a solution to the cube duplication problem. The traditional story with Hippasus of Metapontum is a Neo-Pythagorean myth. According to Knorr, the discovery was a blessing, the study of incommensurables was "a massive project which engaged the best efforts of the most notable fourth century mathematicians: Theodorus, Theaetetus, Archytas and Eudoxus", all Pythagoreans. As far as similar figures were concerned two resolutions were developed.
An earlier approach, formalized by Theaetetus, was based on anthyphairesis, a version of Euclidean algorithm, applied to segments instead of numbers. The algorithm produces a sequence of integers (continued fraction of the ratio in our terms), which for incommensurables does not terminate. However, anthyphaireses of two different ratios could still match. "Such is the definition of the same ratio" writes Aristotle, and goes on to explain how the theory of similar figures can be developed without common measures based on this definition, see Fowler's paper and book. To the extent that Pythagorean "everything is a number" required reduction of geometry to integers, this accomplished it. Theaetetus also classified "magnitudes commensurable in square only", what we call quadratic irrationals, Book X of Elements is based on his work.
However, the anthyphairetic calculations are unwieldy and there is rarely a pattern in the terms, so this approach was abandoned once an alternative emerged. This alternative was due to Eudoxus, who defined two ratios to be equal if they end up on the same side of every integer to integer ratio (Euclid phrases it in terms of multiples). Note how again only equality of ratios is defined, not what a ratio "is". This definition is much handier and allowed Eudoxus, and later Archimedes, to develop a theory of areas and volumes ("method of exhaustion"), that would have been insurmountable with anthyphairesis. Euclid adopted it in Book V of Elements, and redeveloped the theory of similar figures based on it in Book VI, "method of exhaustion" is taken up in Book XII. Dedekind cuts were inspired by Eudoxus's theory of proportion, however unlike Eudoxus Dedekind treats ratios as numbers proper, i.e. conceives of them as separate items, rather than placeholders in a proportion, and defines arithmetical operations on them.
You can see this post on the discovery of the irrationality of the $\sqrt 2$ and this post on the collapse of Pythagoras' school because of ther discovery of irrational numbers.
We can say, simplifying a little bit, that the discovery of the existence of irrational magnitudes leads Greek mathematicians to the withdrawal of the so-called "commensurability assumption", i.e. the (implicit) assumption that :
given two magnitudes, e.g. two segments of lenght $a,b$, it is always possible to find a segment of unit lenght $u$ such that it measures both, i.e. $a=n \times u$ and $b=m \times u$ for suitable integers $n,m$.
From this originated the need to axiomatize geometry, i.e. the need to explicitly lists all the needed assumptions.