Using the formula
$$E ~=~ \frac{\pi^2\hbar^2}{2 m a^2}$$
where $a$ is the length of an infinite potential well. It is apparent that as $a$ get smaller i.e. from a metal to the size of an atom, the energy of ground state of the particle gets larger.
What is the purpose of knowing the value of this ground state energy? How does it help knowing that in a metal the ground state is on the level of ~ femto joules and in an atom is on the order of joules...so and so forth.
One of many good reasons to know it is that you can also use the value you get for a metal, along with the dependence on the number of the energy level (usually "n") and the fact that electrons, as fermions, "pile up" in terms of their energy states, to get an approximation of the energy distribution of electrons in metals. I would second the notion given by @Alfred Centauri that doing canonical problems like this build your intuitions in quantum mechanics, a famously counter-intuitive branch of physics!
As you yourself have illustrated, you gain an understanding of how the energy changes with respect to the parameter $a$. For fixed $a$ you may argue that this ground state energy is just an irrelevant constant. But if $a$ is not fixed (e.g., you cite two different values of $a$ in your example) then this formula does tell you how the energy varies with $a$. Would you have guessed the inverse square dependence on $a$ out of thin air?
