What is meant by “the priests, the men of the surrounding area”?

The phrase surrounding area wants us to think of the local neighbourhood. The men of this corner of the city were (predominantly) priests (recall that Eliashib the high priest also had a home in this area; Nehemiah 3:20). We might be surprised that the priests were living some distance from their place of employment (the temple), but we ought to bear in mind that Jerusalem was not that big. The point Nehemiah is making here is that residents volunteered to repair in their own back yards, something we can certainly understand. In fact, in the remainder of the chapter several of the crews work on the wall near their own houses (Nehemiah 3:23, Nehemiah 3:28, Nehemiah 3:29, Nehemiah 3:30).