Yes, I've exported and self-hosted in the past. I'll sometimes use Sites to mockup something and then I'll download and pass it onto another agent for use in something bigger.

From what I can tell, "Sites" product supports custom domains so it's not an "export" per se but it is a way to map.

https://learn.chatgpt.com/docs/sites?surface=web#connect-a-c...

https://help.openai.com/en/articles/20001339-creating-and-us...