Both idealism and materialism have convincing arguments for me, yet within each argument are refutations of the other. How can I reconcile which to believe when they both seem equally as likely? I've thought that perhaps idealism explains our own subjective worlds, and materialism explains the objective external world, but can both be true when they contain refutations of the other?

I'm not sure what arguments you have in mind for these two views, but I understand, I think, the main feature that raises your question: each implies that the other is false. And yet, as you say, you find that each is equally likely. If each implies that the other is false, then they can't both be true, to answer your last question. One of them, at least, must be false. Perhaps they are both false, if they both rest on a false presupposition, for example. This seems to be the situation with the two views, (a) that the present king of France is bald and (b) that the present kind of France is not bald. They contradict each other, so one, at least, must be false. In fact, it seems they are both false, since there is no present kind of France at all--though I should note that, at one time, more philosophers thought the two views lacked a truth value at all (that is, they were neither true nor false, so on such a view we could say that the two theories you are considering might lack a truth value at all). So...