How do you say it correctly?
- Adult and children stories
- Adults and children stories
- Adults' and children's stories
I'm not sure what you mean: if you said "adult stories" that would probably mean "porn". "Children's stories" would usually mean "stories by children" rather than "stories for children". The best fit for what you seem to mean is
Stories for adults and children
Stories for all ages
I work in children's book publishing, and the standard within the industry would be children's stories, and would refer to stories written for children, usually written by adults.
In the kid's lit world we do use the term "adult books" to refer to non-porn books written for those over 18, but I admit that's not common usage outside this niche.
Adults' and children's stories
That's probably what you mean.