It is straightforward to run an isolated network with TCP/IP, DNS, and web servers. The hard part would be dealing with software that complains/fails if you're not using HTTPS.
In general, you would want an offline copy of the entire software stack (e.g. a Gentoo Linux mirror) so you can patch whatever problems you encounter.
You would have to run your own CA and get everyone to install your root cert on their devices. That what happens already with tech like smart cards and SSL inspection firewalls.
It's all about trusting the connection implicitly.
It would mpst likely come down to infrastructure maintenance capacity, so if we're tallking regional or sub-regional maybe. For example southern california probably has enough industrial capacity that so long as raw materials can be acquired maintence would be relatively simple. But if we're talking scattered individual townships without much intertown services then a BBS would probably be easier and more practical to maintain.
Depends on how dense or spread out it is.
If it's dense and everything is in one place you might not need anything more than just going outside.
If it is spread out though you may want communication methods other than mail and if people already have the computers and existing infrastructure (many places have cables for network and phone lines) to set up a local network then that might be the best option.