“I’ll give you one boulder for a cow.” “Ok, it’s a deal.” “Cool, cool. The boulder over in so-and-so’s field is your now. Pleasure doing business with you.”
This is how gold standard currencies work (or used to work). Most western currencies like the US dollar used to be permanently pegged to a specific value of gold kept in national treasuries (the Bretton-Woods system), and the dollar was meant to be redeemable for this gold. But because of the impracticality of handling and storing actual physical metals actual trade was almost never handled in gold.
In reality, the US financial system already kind of runs like your “dystopian future”, and has done so since 1971. There is no inherent value to a US dollar besides the federal government saying “trust me bro”.