Whatever it is, it avails not—distance avails not, and place avails not.
This is the shortest section of the poem, so Whitman intends to set these words apart. They are one of the climaxes of the poem, where he states his conclusion to question of whether people can be divided from one another in any sense that truly matters: no way.
Neither distance nor time can "avail" or succeed at putting up a wall between individuals.
But in that case, why do we feel so strongly that we are separated from people. Maybe it's all in our heads…