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 a correct usage of "just"much better than the original. You needIt might be colloquial, but it is not common in formal, written English. Typically, one would use a conjunction, andrather than "just"—and probably a semicolon as well {; but, ; however, ; unfortunately ;alas, ; regrettably,}