This is not impossible. This is merely a completely separate connection.
Think about this for a second. All the internet is is a connection between billions if not trillions of computers. Think of it as an intranet of internet. Why would this be impossible?
It wouldn't be anything big like a whole new infrastructure layer, and because of what it would be and because people created it in a modern age the government couldn't walk in like they own the place. Whole new protocols could be targeted for this and experimented with. EVERYTHING could almost act like it was a VPN.
This isn't really all that far from reality, some work just needs to get shoved in.
And unless it's using the existing infrastructure, you'd need an entirely new infrastructure, or tons of wireless devices all bouncing the signal along, adding a lot of ping to things.
Thats one way, and yeah technically TOR is an intranet. However you don't go to google. You don't go to anything. Everything is user made and every user would be a server, much like i2p. Though, instead of having to have a bandwidth cap on an encrypted line, this is full speed to whatever the hell you're doing.
Like an empty room. Fill the room.
I hope someone else like @DeusQain has some ideas.
If it comes to this, it would probably be better to just start over and use only IPv6 and use almost all new infrastructure. If it gets to the point where we can't trust ANY infrastructure anywhere, VPNs probably won't help. I've rolled this idea around a few times, and while it's fun to create a VLAN and assign public IPs to things just to see how badly you can break services, it's not really a practical thing to do right now.
TL;DR - Possible, absolutely. Practical/useful/best course of action? Probably not.
If you're going to want to create a new 'internet' there is a lot we can learn from the shortcomings of the current internet.
A new addressing scheme that names locations instead of interfaces, better solutions to multihoming, implementations of protocols in the session layer that work better for high bandwidth, latency sensitive, and reliable connections.
And for goodness sake, the biggest thing we need to fix is freakin' video streaming. There's things like UDP multicast that exist to make it mind-blowingly efficient, but nothing supports it. If we're starting over we could so totally fix that, and that would make things like IP television and YouTube more efficient.
A lot of vague talk in here. You know, a VPN isn's some magical anonymizer, it's just a way to use the internet as if you were on some other ISP, either local or in a different country. The internet in the USA is pretty terrible for video streaming so it makes sense to use one. No need to start over, just fix your country.
We need the push for more 10 gigabit/s NIC's for consumer hardware for home networking and better support with actual working drivers for Wireless AC NIC's. You can create a base home server with a caching scheme that will cache particular websites with certain data like specific Youtube content, or your Netflix media. This allowing a much faster rate of transfer between your media device and the server.
I've been farting around the last 2 weeks trying to get the BCM4360 Wireless AC Broadcom chipset to work in Linux and have only found one stable configuration that will allow it to work. Kernel 3.13.063 and the bcmwl-kernel-source_6.30.223.141+bdcom-0ubuntu2 source driver, the version 6.30.223.248 driver does not work properly with any kernel version. You also cannot get the 6.30.223.141 version to build with any higher kernel than 3.13.063 because it's not supported.
This is just some personal experience I've had that correlates to this, in that, we have a lot of hurtles to jump both on the consumer side of things and the hardware manufacturer side of things.