I don't think it has to be "believeable" as in what takes place in our world today. However it has to be "plausible" within the world the writer creates. The events that take place for example in movies such as "Star Wars", "Batman", or "Inception" are not (realistic) by any means when we look around us. This is why it's important for a writer to be able "set up" their story very well from the start showing what is "possible" within the world they have created.
What turns people off is when a character is suddenly able to do something or has a power which was not alluded to earlier in the story. If someone is drawn into a story they will overlook a lot of things. (ie) You ever notice Superman is able to change into his costume at a moment's notice. Now we buy into the fact he may be wearing the outfit under his reporter suit (cape and all)....But what about the (RED) boots!? Are there a pair in every phone booth in Metropolis? LOL!
A large part of reading fantasy or going to the movies is to escape "reality". We just want the characters to be "realistic" within the confines of (their world).