Interesting thread that I missed 
I think this discussion needs to be framed in the context and not in isolation. First question is what it would mean to have an unplanned sudden-onset electricity down scenario that covers a larger area (like potentially caused by an Electro Magnetic Pulse (EMP) attack or Coronal Mass Ejection (CME)). It means virtually no more heating in cold areas, no functioning financial system (i.e. no ATM, all non-physical cash payment systems down, no banks, suppliers are not getting paid anymore), logistics systems down, no resupply of supermarkets, all critical services (water supply, sanitation, health services, security services, …) down, medication inaccessible, fuel supply down. Supermarkets emptying within a couple of days (assuming those with physical cash are still served), the general sense of insecurity and uncertainty would put a lot of mental stress on the population. If this happens in a developed country at the level of one medium to larger city, this is probably very costly but overall manageable, especially if trust can be maintained and people see that solutions are in the pipeline. If this is covering a larger scale this would get messy very quickly.
The lower rungs of Maslow’s hierarchy of needs suddenly will become the main focus of everyone.
Very quickly things will almost exclusively turn around securing food, water, shelter, possibly warmth, health needs, security. Some elements from higher levels like communication will also be important to obtain news and stay in touch with family/ friends.
In this context I think there are two issues here that need to be looked at separately:
Can the Cardano ‘system’ be kept operational? I’m not deeply knowledgeable on the technical side of the Cardano hardware/ software system itself, so correct me if I’m wrong here. I did try to get a similar discussion up in the thread The blockchain as a tool, so I won’t repeat things already mentioned there. There would have to be nodes that keep the Cardano system up-and-running. These can probably be running largely outside of the affected area. In the area you’d need enough working contact points that allow you to connect to the Cardano network. This probably needs working mobile stations (smart-phone, computers - did they survive, could they be supplied? - with the adequate software) and network (sub-)stations that can make the link between mobile stations and the external intact network. Not enough of these? Not working well? Not enough bandwidth? Your system is at best crippled, possibly useless. Paper wallets will not solve this as access to the network is required to verify is the ADA it represents is valid (or just a Xerox copy of a dud). With enough foresight, preparedness, adequate set-up of the system and resources, a limited working Cardano network could most likely be set up in this situation, but it comes at an additional cost. Probably best to hitch-hike on, or support, a system that provides a larger functionality to society (e.g. mobile phone network). Don’t underestimate either the difficulties in operating in these kinds of contexts, and getting things up-and-running (and keeping them running). Solar systems and generators are extremely valuable in these kind of contexts, they tend to get legs 
(much of the blaring music on markets in rural Africa is powered by solar panels, I expect that many were imported with another initial use in mind…). So security measures will be needed to ensure they remain operational…
The second question is why use the Cardano network in such a context in the first place? I could imagine that commercial players in the affected area could use Cardano to pay external suppliers B2B. This would probably have to be prepared for before the event though. Trust in the system would be an absolute prerequisite for stakeholders to be willing to use it. Trust in the value would also be a prerequisite, in large-scale disruption I don’t see ADA (or any cryptocurrency for that matter) provide value stability (or even maintaining much value at all).
If ADA is just used as a means of exchange P2P, then we have the solution already in physical cash (I agree, it’s fiat and thus backed by nothing but trust in the issuer). In the scenario mentioned above we’d probably go from electronic currency (knocked out) -> physical cash (used in first days, until people realise that you cannot eat cash) -> barter. Where would ADA come in here? Would the Cardano system be able to provide added value in other things besides exchange of value? Assuming that the Cardano network can be kept up there may be uses. Registration of property, contracts, information, identity, ease of set-up of micro-credit or insurance could come to mind. It depends on the scale of the disruption and how it plays out. But again, trust is what it all comes back down to, if the Cardano network is actively working on these kind of scenarios then there may be added value, but it needs to be built into the preparedness and response system, and society needs to be prepared for the use of the Cardano system before the event to build familiarity and trust…