I'm from Taiwan and I would say "說故事" is perfectly natural to me. We also say "講故事". The differences between these two are subtle in Taiwanese Mandarin, if anything, it has something to do with personal preference. I even hadn't known the former (說故事) would sound weird to some Mandarin speakers until I saw the answers above. In Taiwanese Mandarin, 說 can mean "to tell", "to say" or "to speak". So it's not just a colloquial but formal usage in Taiwanese Mandarin. The dictionary published by 教育部 includes this meaning (to tell). For your information, it comes from ancient Chinese literature. An example in the dictionary dates back to 500 AD.* It is understandable that it became a regional thing after the long-time evolution of the language. By the way, in Taiwanese Hokkien, the second-most-common language in Taiwan, only "講故事" exists. Just on the contrary to Taiwanese Mandarin, 說 hardly means "to tell" in Taiwanese Hokkien and 講 could mean "to tell", "to say" or "to speak". *劉義慶《世說新語．德行》：「有人向張華說此事。」

讲故事中的这个“讲”字和英文中tell的意思是一样的，而“说”，在英文中应该是say，英文中要表达“讲故事”的意思，是绝对不可能说say a story，所以，现在你应该懂为什么用“讲”,而不是“说”了吧

@Justin: I didn’t make myself clear either. I was commenting on Alejandro’s comment, other answers and what you were told. I was shocked that quite a few people think in this way. Sorry about the confusion. To answer your question, yes, both 講故事 and 說故事 are correct to me (speaking of Mandarin) and also to most Taiwanese I think. I elaborated in my answer, but if it’s not clear enough feel free to ask :)

@Yi-Ting: Hi, I didn't make myself clear. I wasn't necessarily saying 說 always meant "to say" and that 講 always translated to "to tell". I have learned other languages in the past so I know that's not how things work. But I did want to ask you, as someone from Taiwan, are 講故事 and 說故事 both correct to you? Thanks.

In my humble opinion, as a native speaker, you could simply say it doesn't sound natural to you and it would be justified. But I think it's kind of pointless to use word-for-word translation to English, and judge a usage based on English logic. 說 does not necessarily mean "to say", and it's not always the case that collocations in English also apply to Mandarin. (Actually it's always not the case. As a native Mandarin speaker, I do have a hard time with English collocations.) Indeed, "To say a story" is weird, but this has nothing to do with whether 說故事 is natural. It reminded me something I came across recently: I'm learning German and I found that "to say" is "sagen" in German (they were the same word which has gone through some consonant transition), but "sagen" can be used both as "to tell" and as "to say". I don't think German speakers would agree that "Ich habe dir gesagt" (literally: I said you) is "weird".