Measuring a binary number with maximum electrical certainty
I'm a novice at electrical circuits, but I've worked out that I can perform this operation using 6 exNors and 5 And gates.


Yes you can (if 6 is the number of bits you want).


  I'm assuming there is some small uncertainty in these logic gates, and that there's a super small chance that these gates can misfire (maybe even so small it's associated with the quantum electrical noise).


The chance is negligible, by which I mean, it doesn't happen. You may as well ask whether there's some small uncertainty in a tennis ball's position, so that there's a super small chance that it will teleport across the room (maybe even so small it's associated with quantum noise). Or the probability that every now and then, your car will go backwards for a moment when you press on the accelerator, because of quantum noise.


  If I wanted to make this system have arbitary precision, what could I do to prevent these errors from /ever/ occuring?


You can't, there's always the possibility that your entire apparatus will quantum-tunnel into the middle of the Milky Way never to be seen again.