The original sentence is okay grammatically, especially in spoken English. It is a bit terse, though. In written English, I would suggest an em-dash, rather than a comma—for stronger contrast (and I would substitute who for that, although there are many who might argue that that is just fine!)
Anyway, one might rephrase it in any of these ways:
I do have friends, but none of them live nearby.
I do have friends, but none who live nearby.
I do have friends—it's just that [the problem is that] none of them live nearby.
The construction you suggested
- I do have friends, just none of them live nearby.
Is not much better than the original. It might be colloquial, but it is not common in formal, written English. Typically, one would use a conjunction, rather than "just"—and probably a semicolon as well {; but, ; however, ; unfortunately ;alas, ; regrettably,}