Ham radio bulletin board software permits a variety of access mechanisms: HF, VHF, microwave, internet, dial up, etc. Good for communications amongst a small group.
The Winlink system allows you to connect by radio to foreign internet servers for email exchanges.
None of the above lets you “browse the net” as you can normally. I guess you could use dial up to an unaffected server but that would be slow and who knows what will happen to phone lines.
What’s the bandwidth for this?
I guess the cheapest is 2-m ham, but what would be the transmission speed for text?
You really don't need that capability for a robust network. What you need is an HF-Twitter system that can interconnect with telephone modem connected PCs. HF bandwidth is very limited but it should be able to support a modest Twitter-like messaging service. Of course you would need a robust mirroring system that is located safely out of the U.S. - a Usenet like system. In fact maybe Usenet would be the solution.
Shutting down the telephone system in the U.S. would be a very drastic step that seems unlikely to me.
Would they shut down the Internet? Yes they would and it would be very easy to do so. Just like they did in Egypt: a few calls to the ISPs and it's over. But taking the phone system down would take down the country. Not going to happen IMHO.