"Badassery" depends entirely on how the story is designed.
Though, I assume the most common source of badassery is the inability to be scared or intimidated, and also coping and overcoming what makes you afraid in an (un/)realistic fashion.
One could consider most current action heroes bad-ass by those standards: but it could also be the result of poor characterization.
I have no better example that the Chief in this case (shows you which reference pools I have).
Master Chief is born and bred into war, and are in many ways a war-machine steeped in human form. He has little characterization, but what little there is hints at a person who's used to putting his neck on the line without whining, and that he knows very well that he is there to fight, kill and persevere. Or die horribly trying.
He has a few quips that hints at a person who despite the horrors of war and the weights of his duty, he can keep a humorous personality, even if it could be considered dark, or even one-dimensional (Halo: when he is told that he needs to blow up the reactors of the Pillar of Autumn. Halo 3: He meets up with Cortana and suggests that he'd mix things up by shooting things).
I've not played Gears of War or God of War, (GoW, and GoW???) so I can't say anything about them, but if we were to look at say... Altair from Assassin's Creed (only played the first one, so little to go by) he starts out as very cocky but also very confident- usual traits of badassery- but he is very early overpowered by a Templar nearly twice his size- which lends itself to an indicator that the Templar is even more badass than him.
The game ends with him being more mature and wise, but I haven't finished the game, so I can't say anything.
My point is: badassery lies in the characterization in the characters involved. I'd easily say that a badass character needs a good premise (story) as much as a believable personality.