How are new 'best performance in the world' devices measured?
If you're designing a circuit that measures some quantity, you'll ideally want to relate that measurement back to some absolute physical quantity. This is how NIST and other national standards bodies create the "standards" used to calibrate laboratory test equipment.

The details of how you might do this depend dramatically on exactly what it is your circuit is measuring.

To give an example, one of the conceptually simplest cases would be if you developed a timer circuit that is meant to accurately produce an output pulse once per second. You could relate the accuracy of your timer circuit to the actual definition of the second: "the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom". That is, you could use an atomic clock built around a cesium cell to produce a 9.192631770 GHz reference signal, then divide that clock down to produce an absolute 1 Hz output signal that you could compare to your new circuit's output.

Most likely, you would keep your atomic clock in very highly controlled environmental conditions, while adjusting the temperature, humidity, etc., around your new circuit over the ranges you want it to operate under, to show that its performance is insensitive to those environmental influences. 

Your example of a new ADC or DAC circuit is more challenging, because a "standard volt" is not easily produced, but must be related to fundamental physical measurements of time and current using the Josephson effect.

Finally, in the worst case, that you are in fact producing a device of the accuracy of the national standards themselves, what you would have to do is convince someone else to independently produce a similar circuit, and then compare the two new devices to be able to put some limits on the errors in either one. My understanding is that several national standards bodies are in fact in the process of doing just that to develop the watt balance as a reference standard for the measurement of mass.