Well, you can use relative pitch. While it's true that 12TET is slightly flat (from A440 we use E659.25 instead of E660), it's a difference of just a couple of cents. That's going to sound "right" to just about everyone. If you're playing a solo piece, even a person with perfect pitch would need to have pretty extreme pitch sensitivity to hear that you're off by one vibration every 1-1/3 seconds without a reference pitch!
The problem isn't really nailing the exact frequency, but hitting a frequency that's in agreement with what's going on around you.
Since you mentioned tuning in the question, I'll work with that - the same principles apply to intonation in performance, but since the notes are generally too short in duration, if you can learn to hear when you're "in tune" with a specific tone, the intonation issues will sort themselves out in time.
Vibrating objects (like violin strings) cause changes in air pressure, which our ears sense, and then our brains sort out into sound. We can graph out the actual air pressure, and for a single pure tone it looks like a sine wave:
But the air pressure on your eardrum at any given moment in time can only be one value. So if you're hearing two sounds at once, the waves combine. If both waves are rising, they cause constructive interference with each other, and the wave amplitude (how high the 'peaks' are) increases. If one wave is rising while the other is falling, the result is destructive interference, and the amplitude decreases.
So if you have two sound sources that aren't quite in agreement, the result looks something like this: .
At points A and C the waves of the two sources are in agreement; at point B the destructive interference makes the total of the two waves cancel out.
We hear these changes in the composite wave as changes in volume. If you have a reference playing A440, and you're playing A436 at the same time, you'll hear the combined result fluctuate in volume four times every second. (Mathematically, the difference in the absolute value of the difference between the two waves is the frequency of the "beats").
When you start hearing those beats, move your finger a bit. If you're getting closer to an exact frequency match, the beats will slow down - at A438 you'll only hear two per second. If you move the wrong way, the frequency of the beats will increase.
Taking this back to the difference between "pure" fifths and 12TET fifths, if you play E660 against E659.25, you will hear one beat every 1-1/3 seconds, the time it takes for the two waves to reach a point of complete opposition. Unless the notes are really, really long, virtually nobody is going to hear that.
As musicians, we strive for perfection. But as musicians, we're also human, and perfection isn't possible. Relative pitch will get you close enough.