Its commonly said that the Pythagoreans were unbalanced by the discovery of the irrationals; since their philosophy was predicated on ratios; ratios of two finite numbers.
Still, it is natural to consider a ratio of two infinite numbers; and most of these will approach an irrational. After all, one easily consider 1,2,3,...; so one might be lead to 1/2, 11/22, 111/222 ...; and one then need only show that some infinite ratios cannot be reduced to finite ones by common techniques: for example (5 x 1111...)/(6 x 1111...) = 5/6. Now, this of course is using imprecise techniques, as far as modern contemporary mathematics is concerned; but different standards of rigor held in antiquity...
Archimedes, much later than the Pythagoreans, had developed a method of exhaustion; a precursor to the calculus.
One might argue this is an outcome of the 'irrational' discovery; but given the apeiron of Anaximander, the boundless; the idea of the infinite as something unbounded was already there.
How historically grounded is the 'standard' narrative of the irrationals and the Pythagoreans? That is their entire philosophy was disrupted:
Pythagoreans preached that all numbers could be expressed as the ratio of integers, and the discovery of irrational numbers is said to have shocked them
Pappus merely says that the knowledge of irrational numbers originated in the Pythagorean school, and that the member who first divulged the secret perished by drowning