I'm finding out that I'm codependent. I avoided this term for a long time because of connotations I had of it with "co-addiction" but really, it is a situation like what you describe - I provide help all the time, and I never ask for or feel like I can accept it. I have so many instances in my past of help coming with strings, or being massively let down, or just being slammed with other people's incompetence that it makes me physically uncomfortable. All of this stems from and reinforces the idea that I have to be responsible because everyone else around me is being messy and useless.
Long story short - ask for help anyway. Even when it feels like shit, you have to get used to it. Like others have said, be specific and direct, and accept the idea that they may say no, and it probably isn't personal. "I need help with..." doesn't work unless it includes, "Can you help me with this?"