No, this most certainly isn't true.
For example, take a meter defined (as SI does) as a fraction of the speed of light.
Now, for every length in the universe to be a rational fraction of this length is most certainly not true, because lengths, under relativity, form a continuum. Any speed is possible, so any length (via contraction) is possible.
There'd be no physical reason all physical processes would be constrained to a subset of rational grid points - things can move freely.
Heck, if all physical quantities are rational, are you claiming velocities only occur as rational numbers? That time is only rational numbers?
It's far more likely that they'd be irrational, and our man made units are the weird things.
Suppose two diagonal corners of a square with rational sides are somehow physically exact.... What is the physical diagonal length?
Oh yeah, irrational.
So no, physics isn't formed from rational numbers.
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.