There are a couple different issues here. As phoog notes, the Pythagoreans loved rational numbers. To them, "irrational numbers" (like various roots of 2 or 3/2) were, well... irrational.
Phoog also notes that you run into a problem when you add a bunch of perfect fifths together. If you use a 3:2 ratio and go through 12 perfect fifths, you theoretically need to add up to seven 2:1 octave ratios for the "circle of fifths" to close. But you don't. No power of 3:2 is going to give you a power of 2:1.
The question presumes that this 3:2 ratio is absolute rather than the 2:1 octave, but that still creates problems in tuning scales.
As to the reason why the Pythagorean scales typically have the particular ratios brought up in the question (9:8 and 256:243), well, it has to do with the ancient Greek method of deriving scales, which was based around perfect fourths, with a 4:3 ratio. The Greeks had many ways of dividing up a fourth into different notes. But one "diatonic" way was to use "whole tones," which were typically tuned to a 9:8 ratio. Why 9:8? Because 9:8 is the difference between a 3:2 perfect fifth and a 4:3 perfect fourth. (Divide 3/2 by 4/3 and you can see this.)
So, when dividing up the perfect fourth, one way of doing it is to use two 9:8 whole tones. But then the last interval you end up with is roughly a semitone. The actual ratio required to complete the 4:3 perfect fourth can be found by taking two whole tones away. So, 4/3 divided by 9/8 divided by 9/8 leaves 256/243. That's where the other ratio comes from.
The question assumes all semitones are of equal size. This was not an assumption in ancient Greece, though it is an assumption in modern equal temperament. Instead, the Greeks would start by tuning the important intervals like a 2:1 octave, 3:2 fifth, and 4:3 fourth. Then they'd fill in the "gaps" as discussed above.
To find the (irrational) ratios for a modern 12-tone equal tempered scale, you'd need to divide the octave (2:1) ratio in 12 parts, hence 21/12:1. That will give you the correct equal tempered semitone ratio, though note that seven of those won't give you an exact 3:2 ratio either, for reasons stated in my first couple paragraphs.