How feasible is it to wire into common household measuring instruments with a solid grasp of electrical engineering?
The question is very open ended as it really does depend on the individual thing you are trying to hack into. Sometimes they have every chip in a nice big DIP package with all the markings plain to see, with the pcb nicely labeled. Other times you get Chip-On-Blob packages with nothing labeled, no accessible test points, no markings, smd chips so small that you can't easily connect a wire to it or bga packages where the pins are underneath the chip body.

The easiest thing to do is open it up, and take a look. Google any part numbers you see, look for projects that access them, look for datasheets. That's the real starting point.

Then there is the question on if you are simply trying to repurpose the item (easier), or if you are trying to read data without affecting how it works (harder). Sometimes you go about it a different way, can't access the raw chip data, how about accessing the lcd's memory and figuring out what is being displayed. There are multiple ways, but you need to know what it has first to be able to come up with them.