From what I remember, the book 'Quantum Mechanics in Simple Matrix Form' explicitly discusses which variables are quantized and which are not, according to stan

From what I remember, the book 'Quantum Mechanics in Simple Matrix Form' explicitly discusses which variables are quantized and which are not, according to standard quantum mechanics. It takes a few chapters to develop the argument. Yes, an abacus stores information, but not an infinite quantity; the information is only a few bits, and it's interpreted that way because we take a very low-precision view of the state of the abacus by dividing the rungs in half, mapping each side to 0 or 1. This is the key; the units of position used for measurement. The object itself can be at any 'true' position (including an irrational position) but no measurement device has enough precision to say for sure; so I think the answer is that objects can take any position, but the fact that measurement-devices are limited makes it a moot point.