To call a character ‘sympathetic’ is one of those descriptors that’s very nearly lost its meaning as people use it as a substitute for ‘likeable’ or ‘realistic’. It is not, however, a dead letter, and it represents an admirable, indeed very nearly mandatory goal for the writer. A character that’s sympathetic is one that we care about because we can sympathize with its shared humanity. It rings true; it comes across on the page as a part of us.
The question of how to do this is an enormous one, and will be a series of posts on its own, but the question of what it actually needs to do is slightly smaller.
The most clear and apparent thing that distinguishes a sympathetic character is one that can safely and plausibly exist within the mind in a way that extends beyond what we actually see them do. A character that we can carry with us later, that we can feel responding to situations, one that we can, and indeed feel compelled to talk about with friends.
More than that, though, we need a character that makes sense within our own paradigms. It has to fit within the remit of our experiences or potential experiences. The golden god on the gleaming chariot is likely not particularly sympathetic – he no doubt has hopes and dreams and concerns, but we can’t imagine what those are, and if you tell us about them it’s like a bear trying to explain to a fish about its knees hurting; we believe him, we understand him, but we have no knees and we cannot comprehend what his life must be like.
Now, what you’re no doubt screaming right now, being the highly excitable and carefully attentive reader that you are, is that a golden god character can and absolutely should exist in fiction, and you’re right. I don’t know why you felt the need to yell about it when I was clearly getting there, but yes, you’re right. Congratulations. Pipe down in the future.
Sympathetic characters are not the be all and end all. It’s entirely reasonable to have characters that are unsympathetic, that exist outside our concept of things, and that expand our minds by having such drastically different experiences and perceptions that they see an entirely different world to us. Just don’t expect the reader to be able to put themselves in that character’s shoes.