An octave (for example from a C to the next C) is a doubling in frequency. In the Western diatonic system, there are 12 notes per octave. (C, C#, D, D#, E, F, F#, G, G#, A, A#, B). Notes are "evenly spaced" within the octave - every note has the same ratio between its frequency and the frequency of the next note. Hence, that ratio is ¹²√2
On a piano you move up an octave by going up by 8 white keys, or 12 semitones (white and black keys). Going up by 4 semitones is called a "major third", which multiplies the frequency of the note by 5/4. If you do three major thirds you get an octave. However, notice that (5/4) multiplied by itself three times is 125/64 which is actually slightly less than 2.
In fact there is no way to tune a piano perfectly - there has to be a compromise in the intervals somewhere. The reason for this is exactly that no rational number (fraction) equals 2 when raised to an integer power.
Just intonation also suffers from harmonic issues when building certain chords, but the tradeoff is that there isn't "beating", or resonant pulsing due to frequency mismatches, since in equal temperament, the notes are slightly detuned in order to fit into the scale, as you've mentioned. Another benefit of just intonation is that it's been observed to be the instinctive intonation used by humans.
There are other tuning systems, which I intentionally avoided discussing because it starts involving LOTS more music theory very quickly. But to quickly describe it: pleasant sounding combinations of notes generally occur at simple ratios. If you look at a simple major chord, like C-E-G, the E would be at 5/4 the frequency of C, and the G would be at 3/2 the frequency of C. However if you tuned a piano like this, it would be specifically anchored to the root note of C, as that's what we're referencing those ratios from. (This would be "the key of C major.") It just so happens that in 12-TET tuning, we get ratios that are very close to these simple fractions, and since the tuning is "equal", it works for any key/root note.