Difficult without a specific case.
Frequently, to find hole layout dimensions for odd shapes, I will use a digital camera or scanner to take an image of the part next to a ruler. Plug the image into a graphics or CAD package (my preference), and use scaling and measuring tools to find the exact placements.
With threaded holes, I've sharpened screws, and screwed them into the object's holes so they slightly protrude. Then press the object onto graph paper, maintaining some sort of XY orientation. The points will leave marks so you can measure the hole positions with a ruler or caliper.
With un-threaded holes I would do the same, except just press a sharpened screw through the hole (s).
I suppose you could make an XY table with threaded rods to move it, and graduated knobs with markings for sub-rotation divisions. 20TPI threaded rods and knobs with 10 radial marks would read out 0.005" resolution. And a small magnifying lens with cross hairs for a reference position.
Ken