Nobody works for a living in the Star Trek Federation since the concept of money does not exist. Without money as a medium of exchange, how does the Federation conduct trade with the other worlds? I cannot imagine how the money-minded Ferengis would want to have any dealings with humans if no money is involved.
As you can see from this image from TNG: The Price, when the Federation deal with non-Federation state actors, they offer something called "Federation Credits". This would appear to be a monetary unit that (presumably) can be exchanged for goods and services.
The first line of the Federation bid reads:
- Lump sum payment of 1,000,000 Federation Credits to be paid ... 100,000 Credits to be paid for annual access"
When trading externally with non-federation entities it's possible that they use valuable minerals, elements, etc, such as Plutonium, Gold, Diamonds, etc.