My use case is very simple. I want to create a chain where peers can store some public data. What is the best way to accomplish that in Substrate?
I think I should implement a custom run time for that, but I'm not sure how to create a transaction sending data. I didn't found anything on that.
You may be looking for something like the system module's remark transaction.
It allows users to submit arbitrary pieces of data and have them attested to by the blockchain. That feature is available in any Substrate based blockchain including the node template.
A good place to start learning how to build custom runtimes and explore your idea more is the Proof of Existence tutorial.