Then on the practical aspect he works at night or in the dark and has technological advantages (as well as ranged non-lethal weapons) over his enemies.
Speculating you could imagine he is pretty good at reading people and their reactions, and so has a rudimentary ability to avoid (not quite dodge) gunfire.
Sure, I think it is unrealistic that in, say, the films he avoids all the gunfire period. But it seems reasonable that he can avoid most of it with these advantages.
Such things are all possible in real life, but I expect you are right - you would end up shot.
It's starting to sound eerily like Anonymous, actually.
"Is he the original Batman?"
"No way, he'd be a hundred years old!"
"Maybe he's immortal..."
The Dark Knight had an interesting take on this with the impostor Batmen in the beginning using guns, and the real Batman showing up to stop them. (There's something to be said about culture and intellectual property in there too, but that's for another day). I'm sure there are plenty of other good examples as well, but my knowledge of superheros is rather sparse as I was unfortunately never that big into comics.
So you got a number of the main points right.
Zero Hedge's Tyler Durden.
"A retired Indian Gorkha soldier recently revisited those glory days when he thwarted 40 robbers, killing three of them and injuring eight others, with his khukuri during a train journey. He is in line to receive three gallantry awards from the Indian government." [http://www.myrepublica.com/portal/index.php?action=news_deta...]