These verbs all describe ways of opposing people and their behavior, especially beliefs, arguments, and claims, but they have distinct meanings.
Rebuff is perhaps the more interesting of these four words—the only one that cannot mean “argue against.” Rebuff means to resist a person’s aggressive behavior—and only the person who is being attacked, or bothered, can rebuff; you cannot rebuff behavior against other people, only yourself. It is often used in reference to women resisting unwanted (sexual or romantic) advances:
She rebuffed the man’s unwanted advances with a stern stare, before walking away and greeting her friends.
One can also rebuff a person using words, such as by saying “go away!” but rebuff does not describe argumentation, and is concerned with behavior, not beliefs.
To rebut is part of debating. In a debate, first one side argues for their belief, then the other side rebuts—responds in opposition to the other side. This could be in a formal debate, or just a debate-like conversation. It’s a fundamental part of the classical argument structure (as opposed to a Rogerian style of argument, for example).
Refute means to argue, successfully, that some idea is incorrect; it does not mean to just argue against:
Although I argued well, I could not refute his claims because I had no proof.
Repudiate is a little different. It does not mean to argue, but rather “to declare that one does not believe in and does not want to be associated with some idea or person”:
As people get older, they often repudiate ideas and people they followed when they were younger.
Galileo was forced to repudiate his claim that the Earth revolves around the sun.
To repudiate normally involves words, but not argumentation.