You can't host either Google Play or Chrome Store, but you can write your own and install apps from them.
There are many other Android appstores around - most phone vendors and carries have one, many large enterprises have their own and there are a number of 3rd party ones.
I don't think there are any other Chrome stores but I suspect that has more to do with the lack of use Chrome apps have.
You can't install a certificate to enable your Chrome to one-click install a user.js from apple.com or Mozilla.org, but you can install extensions from Google.com. That is not an open web.