As others have said -- and probably beaten to death -- a name does not have to make sense. Specifically, a name is not a sentence or even necessarily a grammatically correct fragment of a sentence.
"Elm" of course is a kind of tree. In the context of a place name I assume "barrow" refers to a dirt mound.
This one strikes me as a little bit odd because of the ordering of the words. If someone called a place "Elm Barrow", I'd take that to mean that it was on or near a barrow that was covered or surrounded by elm trees, or at least that had one or two prominent elm trees in the vicinity. That is, "barrow" is a noun and "elm" is used as an adjective describing something about the barrow.
But "Barrow Elm" is backwards. There can't be a barrow sitting on an elm tree.
But as it's a name, maybe whoever made it up just liked the sound better if you switched the two around. Or maybe they are intended to be unrelated: there is a barrow and there are some elms. I don't know how you could know without asking the person.
I just checked a dictionary and I see that "Barrow" is also the name of a river in Ireland. Perhaps the person who named this place was thinking of that river and of an elm tree that grows along that river -- a "Barrow elm".
I see that "barrow" can also mean a castrated pig. The tree belonging to this pig, or where this pig lives? "My pig's tree" seems a funny thing to want to call a place, but as I'd never heard that definition of the word before, I don't know the connotations.