I'm making a statement about all possible experimentally measurable quantities, which as you pointed out have a finite precision and so can be represented as ha

I'm making a statement about all possible experimentally measurable quantities, which as you pointed out have a finite precision and so can be represented as having terminating decimal expansions. This as far as I'm aware this is not a controversial statement. I'm not making any kind of statement about whether space-time is continuous or not. You are modelling spacetime as continuous and seem to be saying this is the only possible option. That's not true; there are a number of different models of how space-time behave at the smallest length scales. Try reading about loop quantum gravity for example; in that model spacetime is taken to be a discrete lattice at the smallest scales. In your model with continuous space-time then it's true that a square of side length 1 would have irrational diagonal of length root 2. In reality we don't currently have experimental evidence to say either way, and it could be that your square of side length 1 has a rational diagonal which approximates root 2.