Turns out, this has been asked before or, more accurately, someone tried to create such a site (even the effort has been deleted from Area 51) and it failed. So...
Worldbuilding.SE legend has it that the idea of Storybuilding.SE was once discussed, but at the time the rules of Stack Exchange were so rigid that such a Stack could never succeed or even graduate from Area 51.
Time has passed. It's been proven that people completely ignore rules and that few people are really interested in enforcing them. Further, Stack Exchange appears to be relaxing its belief that every answer should have the immutable objectivity of solving a mathematics problem. Indeed, since Stack Exchange seems to care very little if we're following any of its rules at all (interventions in the last 5 years have been so rare that I can remember only one), there may be some wiggle room.
And if this Stack has proven anything at all, it's that people have as much fun (or, frankly, even more fun) answering storybuilding questions than they do worldbuilding questions.
Question: Is it time to try and create Storybuilding.SE over at Area 51?
NOTE: Do NOT run over to Area 51 and create Storybuilding.SE. It's a LOT more complicated than that! We'll be required to establish the rules of behavior for that Stack and that deserves more thought than just ringing the proverbial bell. A minimum number of interested participants are required, and if we don't see a lot of support here, it's a waste of time (might even be a negative, setting a precedent) to try to create the Stack. On the other hand, it would be helpful for people interested in this process to visit Area 51 and familiarize themselves with how that service works. And remember, the only people who decide if a Stack graduates Area 51 is Stack Exchange staff.
Issues:
Storybuilding.SE (SB.SE) will be stuck with the same Help Center problems that Worldbuilding (WB.SE) suffers and then some. This means that Storybuilding.SE's creation must include, from the outset, policy-based rule overrides. SE allows communities to override rules, but they don't like it and want to see it minimized.
If SB.SE were created today, there would be an enormous blur between SB.SE and WB.SE. That's a very real problem as it would be best if the two sites conscientiously migrated questions to the appropriate Stack. I wouldn't be surprised if our Stack Exchange Overlords expected us to clearly differentiate between the two Stacks.
Stack Exchange does not want to be perceived as another Reddit or Quora. That means establishing real and practical rules for excluding questions. If the Area 51 participants can't explain what questions are off-topic, SB.SE will likely not be allowed to graduate Area 51.
There is no guarantee that Stack Exchange will allow SB.SE to graduate Area 51. It would be the most subjective Stack they have. It might be the most subjective Stack they ever have.
Finally, it's worth noting that an astute observer might ask, "why don't we just broaden ourselves here at WB.SE so we can answer story-based questions?" There's still a LOT of resistance to story-based questions here. Therefore, I don't believe it's worth even trying to open the doors to storybuilding questions on this Stack. Besides, I can see a clear difference between the question goals, so I believe there's value to having two sites.