My best friend sometimes shuts people out when she's going through a difficult time, for whatever reasons. Once when she was doing this, I backed off and gave her space. Now, for the last week, we haven't talked to each other. When I asked her about it, she said that she was giving me space. So then we started arguing and everything we never talked about between us came up. She told me that its hard to be my friend and what I say upsets her sometimes and she doesn't even feel like trying anymore. I told her that when she spaces out I'm hurt and that I'm tired of having to be the one who tries to fix our problems. Everything boiled over and she told me that when I hadn't talked to her for that week, she was hurt and mad and upset. We tried talking through our issues but she said I was too dependent on her and even if we fixed things between us, it would be different than the super close friendship we had. All I want is my friend back and I have no idea how to try to fix things. I want our super close friendship back but I don't know how to get that when she hates me right now. She won't talk to me anymore and I just need to know the first step to getting her back because I would literally do anything for my best friend.

Just the fact that you say she hates you suggests that you have a sort of 'all or nothing' outlook on things. I see no indication that she hates you, nor would a best friend suddenly hate. Hate is reserved for injustice and war and political oppression and unspeakable acts like child molesting, not family and friends. Anger is a cover up for hurt, usually. Some people do clam up when hurt. Some people won't tell the people they are closest to what's going on with other areas of their lives.
Take the high road: tell her that you miss her, value her, that you will be there, that you will listen to any problem and time anywhere, squeeze her hand and walk away.
And don't try to 'fix' her problems if she comes back. People want someone to listen, not fix. Commiserate, not give advice. Feel with them, not tell them what to do. Sometimes just a 'well we've got each other' is all that is needed.