If I were going to be a "superhero", I would have train and train and train. I would have to be in peak physical condition to even consider being a "superhero". Not to mention I would have to spend thousands of dollars on equipment in order to survive fighting crime. You can't just put an XXL jacket on and make a mask and call yourself a crime fighter. Earn a badge and a gun, then you can tell me you're a crime fighter.
It's possible a man to train himself to an incredible degree. Batman-like feats are fabled and rare, but people have done the things he has done (somewhat). I
could believe with enough money, privacy and free time a man
could become adept at all the things Bruce Wayne is, although it would likely drive him insane due to a sexless, restless, friendless existence. I
could even believe with enough money, resources and a large enough team backing him (a la the Outsiders) a vigilante
could have some impact (though not nearly as much) as Batman has in the comics.
The problem arises though that no one has Bruce Wayne's free time, or the few that do wouldn't be training themselves into oblivion.
There are also more problems. Perhaps Batman lives in a world of colorful, easy to recognize, black and white criminals and lawmen, but we don't. You can't patrol around hoping to get lucky, that you will magically catch criminals unawares. Crime happens behind closed doors and out of view, mostly. You can't go around beating people up and expect the local authorities to "protect your identity" as forensics and good ole' fashion detective work will eventually find you out. Lawyers and DAs will happily prosecute your ass for assault and battery. You can't rely on the fact that criminals are a "supersitious and cowardly lot" because, well, they aren't, and to them you're just an ******* in a costume, and you will get shot.
If Batman were real, he'd be all kinds of crazy, more than likely carry a gun, and we wouldn't be calling him a hero no matter how skilled he was at fighting 'bad guys' (as if there were such a thing).