You can verify gold with a vessel of water, an already verified piece of gold and a set of scales. It's taught in story form to primary age children with the famous quote "Eureka I found it!"
For daily transactions at the current gold price, it would be extremely hard to do this sort of measurement that way with any precision without very specialized equipment. A gram of gold (around $60) is a reasonable daily transaction size and has a volume of 0.05ml. In order to distinguish metals of similar densities, you'd have to be able to measure just a fraction of that volume accurately. Maybe there is some clever system to scale this down, but wasn't obvious to me.
Well one historic way of doing this, was instead of trading directly in minute quantities of gold, people would store larger amounts of gold in a bank and trade contracts that represent certain fractions of what is stored. This was called "currency".
We have never seen a distributed mesh network that worked correctly in the real world at scale. The OLPC project tried doing that but ran into severe technical problems. So it's not necessarily impossible, but I simply don't believe we will have an operating mesh network in that scenario.
1: Cryptocurrencies can still operate over distributed mesh networks or sneaker-nets in a semi-doomsday scenario.
2: With gold, it is difficult to verify its authenticity vs some impostor alloy without the necessary tech/knowledge.
Edit: the bottom line is that in a doomsday scenario, people have little use for bitcoins or shiny yellow rocks.