My suggested definition of the multiple barber scenario:
For every men in the village, he does not shave himself if and only if there exists a man in the village (a barber) who does shave him .
From this definition, if there is only one barber, we will always obtain a contradiction, as in the Russell's original BP.
If, however, there are exactly two barbers, B1 and B2, such that B1 shaves every man in the village and B2 shaves B1, then, contrary to the article, the paradox does not remain. No man shaves himself, and a barber shaves every man in the village.
Likewise if there are no barbers and every man in the village shaves himself, the paradox does not remain (thanks WmE).
There are probably many more such non-paradoxical multiple barber scenarios.