An accessory with a Delusion mounted on it. This is the source of the Fatui Operative's power.
That said, removing it would not detract from said Operative having physical abilities far beyond that of ordinary humans, forged from long years of training and combat experience. As for them, they are well aware that they have sacrificed their past, present, and future upon the altar of the Fatui's ideal paradise, such that the damage they cause themselves on account of using a Delusion's power matters not at all.
They fight to the last when in danger not because they fear death, but because they know that the time and resources to hone them into Operatives were dearly spent.