Usually used in online Role-Playing games HAX Mode refers to when one makes oneself invincible and almost god-like can be seen in other online games or even in real liffe situations as well.
Dude 1: OMG he just blocked my ultimate laser
Dude 2: Duh he's in HAX Mode you idiot
Another Example is
Dude 1: Dammit Uryuu Ishida used Quincy Final Form
Dude 2: Yeah your about to be owned he just went into HAX Mode