We can and constantly do. Any time you have a local area network, like your wifi router, that is essentially a “private internet.” You can have a computer on your private home network that serves up webpages in the same way as a website, and unless you tell your router to allow the outside world to connect, that website will be a private one that only people on your home network can access.
This is very frequently done in business and research contexts; if you run a website which allows remote control and monitoring of your power plant, you are most definitely going to keep that on a private network and not expose it to the internet. Many research institutions will have a private network so their computers and equipment can talk to each other, but no connection to the general internet at all, to keep that private network more secure (this is called being “airgapped”).
Going bigger, North Korea has their own “private internet” called Kwangmyong, which has around 5,000 websites and is the only internet almost anyone in DPRK can access.
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