Why is physical distance not modelled on rational numbers? Physicists use real numbers to model distance or physical line. My intuition is completely neutral on whether to use rational numbers or reals for physical quantities like distance, or temperature etc. Is there any historical or psychological reasons do prefer reals? Or there is some mathematical advantages working with real. Is calculus on rational numbers harder than calculus on reals?
Calculus over the rational numbers is really not so pleasant. For example, the Intermediate Value Theorem, The Maximum Value Theorem, and The Mean Value Theorem all fail for functions continuous on the rationals.

Edit: A better alternative for $\mathbb{Q}$ would be the completion of it with respect to a $p$-adic metric (or the real numbers, of course). Calculus over $\mathbb{Q}_p$ has been studied, see here. See also for $p$-adic quantum mechanics.
The very definition of "physical quantities" is that they are "measurable".  Irrational numbers cannot, by definition, be measurable.  So either our definition of physical quantities is wrong, or incomplete, or all operations on physical quantities must be represented by operations only on rational numbers which must reduce to operations on the reals in the limit of idealistic models of relationships between physical quantities, such as the use of calculus of variations to solve Lagrangians and Hamiltonians.