NOT A TEACHER
GoodTaste, I am replying only because "really" is my favorite word in the English language. In fact, some years ago at another language website, the owner courteously and gently asked me to STOP asking so many questions about the meaning of that word in particular sentences. She pointed out that sometimes "really" can have more than one meaning in a sentence, depending on the context. And sometimes the meaning depends on how much stress (emphasis) you put on the word in speech.
Tony: Am I the handsomest guy in the world?
Mona: I can't really/actually say. (As the other posters have told us, it seems to be "I simply do not know the answer.")
But notice that -- to me, at least -- it could have another meaning if Mona adds one word: "I can't really/actually say that." It could possibly mean something like: "I would be lying if I said that you are the handsomest guy in the world. But you ARE definitely one handsome dude! All of my female friends are crazy about you!"