Handhelds (i.e. VHF/UHF, most likely 144MHz or 440MHz) will only work locally without additional equipment. If you have access to a repeater (which isn't a problem, as most are open) that's linked the internet then you can talk to anyone, either via the internet, or via a repeater (that's also connected to the internet) elsewhere.
HF (1.8 MHz - 30 MHz, in select bands) will propagate around the world if the conditions are right, just like shortwave radio. It's really the same thing, except lower power. That "if the propagation is right" is a big if, however, and also requires a large antenna. I also would not have wanted to set up a portable HF antenna in Boston this afternoon - even if perfectly legal, spreading out wire along the ground and throwing a wire up in a tree doesn't really sound like the best idea right now.
Of course, most of the time it's the HF stuff that's the most fun... and although somewhat reliable, and likely to get you talking to someone, somewhere fairly quickly, it's not going to let you talk to a specific person at a specific time with a lot of reliability, even if you've schedule things in advance.
But of course "access to a repeater" violates the part of "with no centralized infrastructure to fail" in the parent's post.
But one also has to distinguish old-school repeaters covering a medium-diameter area from the worldwide, internet-connected relays (such as IRLP). Whereas the former are easily put up freestanding on an amateur budget, and hence will stay in operation independent of all the other infrastructure, the internet-bound systems will cease operation once the internet links break down.
There are some pretty small HF transceivers, like the Elecraft KX3.
I'm guessing that most people that are serious about making contacts across the world when "the big one" hits have mobile rigs and lots of charged batteries, and there are many capable mobile HF transceivers on the market.
There are even smaller homebrew QRP (low power) transceivers that will easily fit in an altoids tin (minus the battery), although I'd love a KX3. You need to know morse code for those to be practical, however. I'd also be, uh, slightly hesitant to bring a circuit board in an altoids tin through a TSA checkpoint.
And HF antennas are still going to be large, no matter what you do. Yeah, with enough loading coils you can get the size down a bit, but then you compromise efficiency. Physics is working against you for small HF antennas.
People seem pretty happy with their Buddipoles and Buddisticks. Those are decent-sized antennas that collapse down to something that fits in your carry-on.
HF (1.8 MHz - 30 MHz, in select bands) will propagate around the world if the conditions are right, just like shortwave radio. It's really the same thing, except lower power. That "if the propagation is right" is a big if, however, and also requires a large antenna. I also would not have wanted to set up a portable HF antenna in Boston this afternoon - even if perfectly legal, spreading out wire along the ground and throwing a wire up in a tree doesn't really sound like the best idea right now.
Of course, most of the time it's the HF stuff that's the most fun... and although somewhat reliable, and likely to get you talking to someone, somewhere fairly quickly, it's not going to let you talk to a specific person at a specific time with a lot of reliability, even if you've schedule things in advance.