Consider these two cases:
1) Let $I_n$ be the closed interval $[\frac{1}{n}, 1]$. Then $\bigcup_{n = 1}^{\infty} I_n = (0,1]$.
2) Take the two sets $y = e^x$ and $y = 0$ (the $x$-axis) in $\mathbb{R^2}$. Then it is possible that the Euclidean metric/distance between them is $0$.
In the first case, we didn't include the $0$ even though $\frac{1}{n}$ gets very close to $0$.
But in the second case, $y = e^x$ gets very close to the $x$-axis, and even though it doesn't actually reach the $x$-axis, we say that $0$ is a possible distance as if they actually touch.
How do you know when to actually assume the value that some things get close to, and when not to? Like why isn't it $[0,1]$ for the first case? Or why isn't the shortest distance some infinitesimally small distance greater than $0$ for the second case?
The phrase The shortest possible Euclidean metric/distance between $Thing$ and $Thing$ isn't defined in your question and isn't mathematically precise. But what it almost certainly means is if $D$ is the set of all such distances, the "shortest distance" is the infinium (also written as inf) or greatest lower bound (also written as glb) of $D$.
Whether or not $D$ contains its inf (or glb) is another matter.
For example, what's the inf of the half open interval $(0,1]$? It is $0$ even though $0$ is not a member of the set $(0,1]$.
Hence your skepticism is well-founded: $0$ is not a distance between the graph of $y = e^{-x}$ and the $x$-axis that is ever attained. The inf of the set of distances is not a member of the set of distances.