What is the precision of Gerber files?

Precision of Gerber files is settable in the Manufacturing Outputs:

The default precision is 3 for digits before desimal point and 5 after the decimal point.

The limit of the values that can be set for precision is 6 i.e. nnn.dddddd which should be more than adequate for anyone.

Be careful putting the value of 3 and less for decimal precision might be too small, especially if copper pour is used in the design (it can overlap the copper on to tracks) 

The best is to check with your PCB manufacturer what precision is required.