The maximum theoretical precision of an analog computer is limited by the charge of an electron, 10^-19 coulombs. A normal analog computer runs at a few milliamps, for a second max. So a max theoretical precision of 10^16, or 53 bits. This is the same as a double precision (64-bit) float. I believe 80-bit floats are standard in desktop computers.
In practice, just getting a good 24-bit ADC is expensive, and 12-bit or 16-bit ADCs are way more common. Analog computers aren't solving anything that can't be done faster by digitally simulating an analog computer.