You still need an IP address. You can build your own network on top of point-to-point layer-2 connections, which have no central authority, but it won't be reachable from the Internet.

BTW: anyone interested in this should join DN42, which is an alternative central authority, and does more-or-less this. Although 99.9% of DN42 links are internet VPNs because that's cheaper, physical links are also accepted because they're cooler.

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