I don't think it makes much sense to go smaller than the planck length (10^-35m) because probing smaller distances would create black holes. So in a sense position is quantized according to current understanding of gravity and QFT.
I guess you can say that, but I think what I'm saying is a pretty standard position in QM:
https://physics.stackexchange.com/questions/32665/why-positi...
Is space-time known not to be quantized? Or is anything?
Does a slide rule or an abacus store information, or does it just "sit there"?
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.
Position isn't quantized; energy is.
An object having a position described by an irrational number does not store infinite information. The object isn't storing anything at all; it's just sitting at that location.
The person doing the decimal expansion has to store an infinite amount of information. What if the person just uses an infinite-series expression?
Another question - what if I just shift my reference frame to be one irrational unit offset from yours? Does the information encoded by object-positions in my reference frame now change?