During many of the explorations into the Star Trek franchise, Earth has been shown as a shining beacon of human creativity and ingenuity, the pinnacle of what humans can achieve when everyone bands together and looks past their differences. Gene Roddenberry created this world specifically as an example of what a culture could do when issues of race, sexuality, and more, were not a discriminating factor. He also presented a civilization where money, something in modern society that tends to be heralded as the building block of everything, is obsolete.
It's a nice change, this wonderful prediction of humanity's future, distancing itself from the dystopian projections common within a lot of science fiction. The members of the United Federation of Planets live unbound by the confines of being obliged to earn enough money to simply survive. Instead, everyone in their society is given the necessary things required in order to live. Food and water are provided simply by asking a replicator for something — even things like clothes can be replicated in a blink of an eye. Who needs a walk-in wardrobe when one can download any item of clothing in a second, like crafting in a video game? This does raise the question, however: why, if one gets nothing from it, would anyone have a job?
Explaining this comes, in part, directly from one of the audience's most beloved Starfleet captains. Jean-Luc Picard, coming back right now in Picard's second season, states that once world hunger was solved, humans moved past their infant need for possessions. What motivates people in this utopia is not money, but prestige and admiration from those around oneself, as well as an inherent desire to be the best person possible. There is a hunger for truth and justice, as well as knowledge. This is not to say this is a perfect system. Problems challenging modern society are still present, such as an obsession with power or influence — it’s just that these issues are no longer tied to money.
Jobs, then, are things people do because they actually want to, rather than laboring to barely be able to scrape a living. It does pose the question, however, about all the jobs that people don't want to do. It's probably fair to say that jobs, while they make for a valuable, honest living, probably aren't most people's first choice, especially when all monetary incentive has been taken away; for example, a sewer worker, garbage collector, or god forbid, Aperture desk worker. While not addressed specifically within the franchise, with the Federation's advancements, it’s probable that jobs such as these have been automated. This removes the need for someone to actually commit to doing the jobs that are unfortunately frowned upon in today's society. What’s left, then, is "work" that resembles play, activities that people do because they are passionate about it. Such occupations range from scientists to chefs, Starfleet captains to gardeners.
Currency still exists in this universe, however, as the majority of other species outside the confines of United Earth still use some form of money. The Ferengi worship currency like religion, and money is shown to be in use even on Starfleet maintained vessels such as on Deep Space 9, where the stalls and shop owners aboard trade and barter outside the utopian vision. Starfleet itself even uses a form of currency when dealing with these outside species, but in a much more primitive form, trading things like lithium for things they need. While they themselves do not adhere to a society governed by currency, this does not mean they do not understand its importance outside their little bubble.
This of course comes, in part, from a place of privilege. It is all well and good to say this is the ideal model of society, but it's in an envisioned future where resources appear to be abundant, and material wealth has grown to the point where physical possessions don’t carry the same meaning. The only thing that people can amass is status, intelligence, talent, and the idea of building a legacy — a driving factor for many Starfleet members. This idealized future where money is obsolete is also something that completely relies on political and social consensus, not the technological advancements that can support it. The most important aspect is this future vision that humans are good. Otherwise, society could easily (and arguably, more likely) have taken a turn for the worse, arbitrarily charging for the use of replicators, and simply fueling capitalist mentality and greed, becoming more like the Ferengi than Federation humans. It is thought that the guiding influence of the Vucans helped. We already see this in the world today. A prime example is a reluctance to move away from fossil fuels despite the technology being available for electric cars, simply due to a fear of losing the massive profit margin gained from the fossil fuel industry.
The moneyless society presented in Star Trek is a paradise, where people are free to do what they truly want to do without the fear and pressure of having to provide the bare necessities humans need to survive. It’s a wonderful idea and one that could very well be a possibility in the near future. Modern society could take a leaf from the Federation's book, and focus on personal growth and development rather than material wealth and the need to amass the biggest bank account possible.