I'd like to see the Fantastic Four as NASA astronauts that were lost in space/time decades ago, and suddenly reappear in modern day space. That would keep their space race sense of urgency and explain where they've been without having to provide an origin. Maybe the Silver Surfer plays a part in this. Perhaps Galactus is coming.
The X-Men can just be introduced as something happening now. Maybe mutants have been something of urban legend, but now with social media and the internet it is getting harder and harder to hide. Xaiver offers a sanctuary for those who can't seem to get by in society due to mutations/gifts.
No need for any big multiversal crisis, just introduce these characters into the ongoing narrative of the world that has been created thus far.