I'm a bit confused, when you run the machine do all 24 parts get measured (batch)? If so you need 3 machine runs, each run consists of 24 parts. OR is it done one-at-a-time (in which case you get 72 runs?
Why do you need a crossed design? Are you suspicious of interactions between pats and locations? Why?
You could try a sampling approach. For example, do you have any hypotheses as to why there would be differences in location within machine? Is there any historical data that points to certain locations having issues (or not)? If you don't have any hypotheses regarding why they are different, you could randomly select a subset of locations, though I'm not sure that would help as you still need repeats. You could reduce the number of repeats to 2 to start and look at the data and see if you need more data.
I've attached a data table for data entry. Again you could do the first 2 repeats and then analyze.
Remember it is critical the parts chosen for the study represent the variation you want the measurement system to be able to discriminate.
"All models are wrong, some are useful" G.E.P. Box