The narrative choices made by Neon Genesis Evangelion rarely have happy outcomes — the world has partially ended, and mysterious creatures known as Angels keep trying to complete the apocalypse. Their own issues burden nearly every single character in the series in addition to the larger problems faced by humanity as a whole.
And somehow, in the midst of all this nihilism, some manage to keep their head held high and almost always have a smile on their face because they know that what people need more than a savior is a helper. Not all of them are such lovely characters, though, but that's necessary to provide a motivational contrast in the story.
10 Gendo Ikari
Gendo is the classic trope of the absent father, except a lot worse. He uses his son, an immature, innocent boy, to fight terrifying monsters for the sole purpose of getting his wife back (a logic that fails him, and deservedly so.)
Superficially, though, Gendo can be considered a hero, since he does really work to protect the world from the attacking Angels, but this pales compared to the way he treats Shinji, Ritsuko, as well as the latter's mother, Naoko.
9 Asuka Langley Sohryu
Asuka is a lost child, but this doesn't (and shouldn't) excuse her borderline-abusive behavior towards everyone, but mainly Shinji and Rei.
Besides, she is a pain to deal with because she refuses to accept weakness; an ironic scenario as this is, actually, a manifestation of her own emotional turmoil. Speaking of which, one of the few times that audiences empathize with Asuka is when she weeps in her sleep, dreaming about her mother.
8 Toji Suzuhara
Toji is a great kid, but only after he accepts his faults and apologizes for harassing the new boy at school. Until then, his aggression has no rhyme nor reason to it, making him a classic bully.
In his defense, however, his anger towards Shinji reflects what happens to his sister as a result of one of the Eva-Angel battles, but this rage is misplaced. Unfortunately, poor Toji is left in a medically severe condition when Gendo indirectly forces Unit-o1 to attack his Eva.
7 Shinji Ikari
Although calling him a hero would be a bit of a stretch, Shinji is the protagonist of the story, as the decision for humanity's future rests in his inexperienced hands. For the most part, though, he's preoccupied with his many internal conflicts to care about the situation's reality truly.
This results in him making decisions that work in his short-term favor but go against what everyone else expects him to do. Regardless, Shinji tries his best to look ahead and not dwell on his failed relationships.
6 Ryoji Kaji
Ryoji Kaji's professional and political motivations aside, he is a genuinely lovely person to be around, as annoying as he can be to Ritsuko and Misato.
He hasn't given up, considering that he enthusiastically grows a little garden filled with watermelon plants, indicating his fervent desire to see life bloom. Further, he becomes a great mentor for the child-pilots (mostly Shinji, who looks up to him as an idol.)
5 Ritsuko Akagi
For all the trauma she's been through, Ritsuko is a surprisingly well-balanced individual, regularly acting as a substitute mother for Misato (who is more childlike than some of the actual children.)
Her affair with Gendo Ikari blinds her to the man's manipulative habits. Still, the second she has her epiphany, she ravages a large segment of the core mainframe to show her displeasure. Sadly, Ritsuko meets her end at the hands of the very person she despises the most.
4 Kaworu Nagisa
Even though Kaworu is an Angel, he is the nearest thing Shinji has to a real companion (and, possibly, more.) He loves people's notions, whether he likes them or not, and is the first "villain" who has exhibited some comprehension of human suffering.
Kaworu considers music to be the greatest triumph of all Lilin, as shown by his constant humming of the Ninth Symphony by Beethoven: the lyrical version, An Die Freude, is the background score his final attempt at bringing about the Third Impact. This is extra relevant because it suggests that he at least partially accepts the Human Instrumentality Project's nature.
3 Rei Ayanami
Rei's general personality is all doom and gloom (evidenced by the ambiance of her room and her absolute lack of facial expression in nearly every scene.) Ironically, this is what endears her to the audience; she is seen as a martyr to the cause, unwilling but unable to express her unwillingness, so to speak.
Rei knows, with absolute certainty, that her life only has value in terms of her role as an Eva pilot, but this doesn't mean that she doesn't know how to find happiness on a personal level.
2 Misato Katsuragi
Misato is bubbly and pleasant, almost all the time, anyway. She can't really make food and lives off several gallons of beer a week, but her guardianship of Shinji and Asuka shows that she's one of the most caring characters in the narrative.
On a deeper level, her emotions are shown to be quite murky, especially when they regard her ancient romance with Ryoji, but she never allows this to affect her life or her work. In fact, Misato is the only person who insists that the children be treated as human beings and not as disposable toys for NERV.
1 Yui Ikari
Yui is hands down the most compassionate person in Neon Genesis Evangelion. One might even argue that she's the greatest savior the world has ever seen in the entire history of civilization, and they would not be wrong.
When she was alive, she was an incredible mother to Shinji and (allegedly) an amazing wife, but her "death" is what makes her so important. In a kind of nebulous spiritual form, Yui exists within the techno-organic framework of Eva-01, having sacrificed her physical form to ensure the survival of her only child.