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The world seems to be moving this way, so how would Romans in the classic era have translated the phrase "cashless society". I am ignoring the fact that they didn't use paper money for simplicity.
My first thought was something like Societas Penuniarum Sine Chartas, or "a society that has money without paper", but I'm not sure that would really make any sense because it doesn't convey that the money is actually made of paper. I'm also probably didn't decline this correctly. :)
The Romans had a word for hard cash, as opposed to money or wealth in general: nummi.
One could therefore say: Civitas nummorum expers.
I think it would not be lost on the Romans that this would not mean sine opibus et divitiis, and one could still transfer wealth somehow in such a society; after all, they did have bankers and kept books (tabulae).
Perhaps you could also say things like: Sine nummis emere, obsonari – shop (groceries) without cash, etc.
When first introducing the concept in a Latin text, you probably need to explain what is meant. Bear in mind that while nummus may principally refer to cash, I cannot think of a way to say "cashless" that would probably not mean "poor" to a Roman — or many other people. Therefore context and explanation is needed to make the concept understandable, unlike with the established English word "cashless". But once such explanation is made and you want to continue to refer to the idea in the text, a more succinct expression is needed. With ambiguity dispelled by context, you can go for something compact and idiomatic.
I find the use of a privative adjective to be an idiomatic choice for this purpose. Just like exsanguis and exanimus are deprived of blood and life, respectively, you could coin1 the adjective enummus for something deprived of nummus. (Before N the prefix should be e- rather than ex-.) A similar but somewhat cumbersome alternative is nummatus ("furnished with money", an attested adjective) with a suitable prefix like in-. I would go with civitas enumma or civitas innummata.
There are different options for the prefix. I chose in- to signify negation (it can refer to both negation an "in"), and the prefix is used this way with adjectives and participles. I think e- and de- could be possible, too. Using e(x)- as prefix (in either enumma or enummata) seems to imply that there once was cash but it has been taken away, whereas in- only states that there is no cash. I chose in- with nummata because the option with e- was already available with the more compact enumma.
1 Only an hour after writing the answer do I realize how suitable the word "coin" is. This time the pun was entirely unintentional.
