PDEs of which the solutions have sharp boundaries pose problems that go beyond being able to represent the solution in floating point. This is especially true when solutions have a certain physical meaning, e.g., a density (that per se cannnot be smaller than 0). Consider, for example,
$$
-\varepsilon \Delta u + u = 0 \text{ on } \Omega,\\
u = 1 \text{ on } \partial\Omega.
$$
The exact solution is pointwise positive, but when discretizing with Finite Elements, the may no longer be true. A similar question came up recently.
That said, there is no scaling of variables or domains that removes this difficulty.
One sensible reason for scaling out physical quantities is to have the actual equation as simple as possible, i.e., as few parameters as possible involved (that you'd have to play around with). For example, instead of looking for the solution $u_{\alpha}$ of
$$
-\alpha^2\Delta u = f_\alpha \text{ on } \alpha\Omega
$$
(with some boundary conditions) for many different $\alpha$, you could just go ahead and find the solution $u_1$ of
$$
-\Delta u = f \text{ on } \Omega.
$$
You deduce immediately that $u_{\alpha}(\mathbf{x}):=u_1(\mathbf{x}/\alpha)$ for any $\alpha$. This shows that scaling out the parameter $\alpha$ doesn't really influence the behavior of the solution at all -- a fact that is quite obvious here, but may be quite difficult to see in other cases. You can construct quite complex examples with more parameters in the same way, e.g., Navier-Stokes and its nondimensionalization into the Reynolds-number-formulation.
