EDIT: I suppose it's best to say this before my rant. I laughed heartily at this one, without context Shinji is hilariously whiny.
Original Post: Shinji is a ludicrously flawed character. Self-centred, nihilistic and whimpy... he is not meant to be a hero.
MASSIVE SPOILERS FROM HERE ON:
He's also only 14, has incredibly poor social skills (possibly due to not having parents for most of his life) cripplingly low self-confidence, is bunked up with an Egotistical Narcissist (Auska) and a stoic human doll (Rei, who also turns out to be a [mass produced] clone of his long dead mother and more loved by his father then himself), is forced to pilot a living machine that translates physical damage to real pain on the part of the pilot and randomly transforms in a terrifying monster that, in turn, drives him crazy.
He refuses to kill a close friend (who mid series is revealed to be a new pilot) who's EVA had become infected by a new type of Angel and his father, the soulless bastard, switches Shinjis EVA to a dummy plug (kind of like an auto-pilot) and forces him to watch as his own EVA rips the other EVA into pieces and proceeds to crush the pilots capsule with his friend inside.
Towards the end, he is forced to kill another friend, one who he really felt a connection to, as it turned out he was an angel. This causes him to snap. The betrayal and simultaneous loss just pushes him over the edge and this is what leads up to his fucked up comma wanking scene (the implied sexual frustration, the end of days fast approaching, the complete distortion of what life is worth... yeah he's a little fucked up). No one mentions, though, that Auska ended up in a comma due to starving herself in a bout of insanity (mostly self-inflicted due to aforementioned narcissism)... Shinji outlasts Auska to the breaking point despite having a far worse lot in life and far more pressure to perform.
In the original series rebooted finale (I don't know if this is revised in the later films), shinji actually initiates the apocalypse and ends the world, though I'm kind of confused at how he got to that point (haven't watched it in a while). He didn't initially intend to do it, but after some trippy montages the world ends and shinji seems happier.
This is not meant to be a typical mecha series.
What's funny is that taking all that occurs throughout the series into account, shinji is ultimately the least fucked up and jaded person there. His Dad is a complete nutjob wrapped in megalomania and sprinkled with dogmatism, Auska is an appendage short of fucking herself, Rei (in the begining of the series) is the 2nd of what amounts to a line of emotionless clones and is also somehow significant to the end of the world (forgot her role in it), Misato attempts at "helping" shinji do more damage then they do good (she at one point offers herself physically to him to save the world... at which point shinji ends it).
I can keep going but I start running short of names to associate with the descriptions... even small part characters like his friends in school are fucking mad. They treat these apocalyptic monsters like something from an anime (I know, meta ain't it).
I don't know about others, but I actually found his character more believable then virtually any other mecha series protagonist ever made.