It's certainly not impossible to get a pure interval. THe fifth you get out of a well tuned piano won't be one. It'll be close but.
The fifth of the fifth isn't where the problem is noticed. The fifth is not half an octave so we shouldn't expect two fifths to make an octave. If you tune a perfect fifth (it's a wee tiny bit sharp of your piano fifth) from a C and call it a G, then tune a fifth from that to a D and so on, you'll get the famous circle of fifths:
C-G-D-A-E-B-F#-C#-G#-Eb-Bb-F-C
but the C you end up with won't be the same C you started with. If you read the page wofa posted you'll see that the pure interval fifth is 702 cents (Don't panic folks, one cent is one one-hundredth of a half step. That's the half step on your well tuned piano.) Add up the seven half steps in the fifth and you get only 700 cents. This is two cents flat of the pure interval fifth. Add up all those two centses in the circle of fifths and you'll see that the C you get when you're done with piling all these pure interval fifths on top of each other is 24 cents sharp of the C you started out with. Oh, oops. This is called the Pythagorean comma and it's not a problem as long as you're only playing in one key with no accidentals, but as soon as you want to modulate or have an instrument that will play in any key (such as a piano) you're in big trouble. It took us a couple of hundred years to get this all settled out. We got a lot of compromises along the way that left us with the idea that different keys have different characters. They used to even as recently as Mozart's time. If they still do, it is merely because composers think they do and write music accordingly.
There. Din't even sum no squares.
