What exactly does a "source-free region" mean in the context of E&M? I was reading the answers to this question: How can the electric and magnetic fields be non-zero in the absence of sources?
and the impression I got is that the E field is only required to be zero if there are no sources ANYWHERE.  That makes sense, since the EM force has an infinite range, but, in that case, what does it even MEAN to have a source-free region, since there's always going to be be some power source SOMEWHERE?  I would have thought it meant sufficiently far away from any power sources so that any influence from such sources is negligible.  But if that were the case, then shouldn't the curl of the E field also be so small as to be negligible?
In other words, the crux of the OP's confusion in the linked question is: if there's no power source, then shouldn't the EM field be zero everywhere in the region?  The answers, if I'm understanding them correctly, mainly just say that that it depends on the boundary conditions, as there could be sources elsewhere, which leads to my question: If there are sources close enough to the region to have a non-negligible effect, why are we calling it a source-free region at all?  And if there aren't sources close enough to have a non-negligible effect, for practical purposes, shouldn't both fields just be zero?
 A: The expression $sources$ doesn't refer to a generic cause but to the pointwise and instantaneous presence of inhomogeneous terms in the field equations (Maxwell equation). Such inhomogeneous terms are the charge ($\rho({\bf r})$) and current  (${\bf j}({\bf r})$) density. Their presence in a given region of interest and at a given time means that there is at least a subregion where one or both functions are different from zero at that time.
The answer to the question

If there are sources close enough to the region to have a non-negligible effect, why are we calling it a source-free region at all?

is that a source-free region, in a given interval of time, is a region where
\begin{align}
\rho({\bf r},t)&=0\\
{\bf j}({\bf r},t)&=0.
\end{align}
Therefore, even though there could be some important source term outside, as long as charge density and current density are zero in the region of interest, there is no source term in the equations. This justifies naming such a situation as source-free.
However, the absence of sources in the region of interest is insufficient to deduce that the electromagnetic field is zero.
Due to the nature of partial differential equations of Maxwell equations, non-zero fields may exist in a source-free region at a time $t$ as a consequence of the presence of sources outside the region of interest and/or at times $t'<t$. Such possibilities are encoded in the initial and boundary conditions for the fields.
Therefore, the answer to the question

And if there aren't sources close enough to have a non-negligible effect, for practical purposes, shouldn't both fields just be zero?

is less straightforward: the fields are zero (or vanishing small) if there aren't sources close enough and if, in the spatiotemporal region which influences the region of interest from time $t$ on, the fields are zero (or vanishing small).
Even very far from any source term, a region with zero fields may be along a strong electromagnetic pulse trajectory. As soon as the pulse enters the region of interest, the electric and magnetic fields differ significantly from zero, even if there is no source inside. In such a situation, the sources of the pulse may be very far away, or even completely disappeared, at the observation time $t$.
A: “Source free” in this context means no currents and no charges. It does not mean no energy. Often the word vacuum is used to describe such solutions although an actual vacuum is not necessary.
If you are using the macroscopic equations then “source free” would also mean no polarized or magnetized material.
