Don't lie--it's wrong and it will probably make things worse in the long run.
State firmly and politely that you don't want to talk about it. You can also say "I'm fine," which I think in this context is socially understood to mean "I'm not having a problem I want you to help me with," which is certainly your prerogative. EDIT to clarify, I mean a casual "Oh, I'm fine," not a defensive one.
If it's not as a result of abuse, you could reassure people if they're concerned about that--"I don't want to talk about it, but don't worry; I'm not being abused."
If you are being abused, please reach out for help to something like the National Domestic Violence Hotline, a local shelter, or a trusted friend or family member. At work, keep saying "I don't want to talk about it, but thank you for your concern" and then change the subject.
I don't think this is going to affect your reputation--most likely causes don't reflect badly on you. If other people are prying into something you've already said you don't want to talk about, they are being rude and you are not. Change the subject or walk away.
EDIT (to deal with the more general concern about office gossip): If people are determined to talk, then (depending on you/your work persona and your office culture) you could try distracting them by telling everyone a different ridiculous story, such as "I was defending a child from a vicious T-Rex," "I was punched by a KGB agent while escaping from the Parisian catacombs," and "I was just walking down the street and a bald eagle flew into my face." Make them as elaborate as possible, and tell them with a straight face. That way, the chat around the water cooler moves from the actual cause of your black eye to the funny stories you've been telling about it. If you're being light-hearted about it, I think that will alleviate people's concerns about domestic violence and, if they still speculate, make them more likely to think it was an embarrassing but low-key accident.
A simple "Ha ha, you should have seen the other guy!" will probably suffice, and is also less trouble.