It's important to note that Scooby-Doo is not one unified continuity and that how they got financed if at all. It's also worth noting that the show has generally gone back and forth focusing on mystery and comedy (especially in the 70s and 80s) and was somewhat revived as a mystery with side of comedy in the late 90s. Usually the post 90s is shown that the original What's New Scooby-Doo and First Scrappy-Doo seasons are somewhat in the same continuity but this doesn't include the 70s and 80s especially if the series featued part of the five heroes (The Scrappy-Doo era is often less discussed, but early episodes of him did feature mysteries that were in line with original series). Generally the Scooby-Doo movies, and the adventures featuring Only Shaggy, Velma (dropped later) Scooby and Scrappy are not cannon, nor is a "Pup Named Scooby-Doo" though it was considered better.
Scooby-Doo was revived in 1998 with the direct to video release "Scooby-Doo on Zombie Island" which billed that "this time the Monsters are Real" (despite most of the comedy films already doing that... though this was more inline with classic Scooby-Doo) and featured the gang as adults in their 20s with Fred and Daphne working on a travel show called "Coast to Coast with Daphene Blake" with Daphne staring and Fred Producing. The intro scene reveals that by this point, the Mystery Inc gang known well enough for their mystery solving that an Operah like talk show host discusses their adventures. At the same time, Shaggy is a recently fired customs (for eating all the contraband) agent with Scooby as his canine partner and Velma owns a book shop and Fred hires the three on as part of the tvshow's research staff as the new season will focus on "Haunted America", thus explaining the set up for the later direct to video films and the "What's New Scooby-Doo tv series" and featured some characters from the films coming into the show (Notably the band "The Hex Girls" first introduced in the second film "The Witches Ghost" and were fan favorites).
From this, there were a few consistent elements of financial sources:
Daphne, Shaggy, and Fred were all from well off families while Velma's seemed to be a bit more working class, fitting her more down to earth characterization. While Daphne and Fred's family were proud of their child, Shaggy's laid back attitude was a source of embaressment and his family was more supportive of his activities because it meant not having him around the house. No sources of income were depicted though all wealthy families were depicted as "old money".
The Mystery Machine was Fred's car, and Scooby-Doo was Shaggy's dog. In "A Pup Named Scooby-Doo, Scooby's parents were wealthy dogs as well. Although this was dropped, it's possible that his parents played famous TV show dogs (ala Lassie) and Shaggy's family got Scooby from a liter for Shaggy (personal theory only). Another good theory is that Scooby's family paid them a hefty sum to put up with Scrappy.
The town they grew up in was most likely on the U.S. South East and was a summer vacation town as it had a beaches (with Yacht club), swamps, farms, and one original episode depicted a Confederate Mansion. It is possible that they traveled over a wider area and an alternative location could be California as prospecting history and Asian immigrants were depicted in the original series and the beaches were not in line with those on the East Coast. San Francisco was one of the earliest real world places featured and when given, the town's name is said to be "Coolsville".
From 1998 revival onward, they seem to have had a series of jobs and may be working for a television program or journalist piece. The gang's exploits are known to the wider public as an Operah like figure was more interested in Daphene's stories about the good ol' days then the TV show she was plugging, and Ben Ravenscroft, a famous horror writer does seek them out specifically in another movie. The following films didn't use this angle nor did the "What's New" series, though both depicted much wider international traveling which suggest this could be there funding source.
From here, the series returned breifly to a Shaggy and Scooby only comedy before getting rebooted into the much praised Scooby-Doo: Mystery Inc. which actually filled in alot of gap economic information. This time, the series was explicitly set in the beach town Coolsville, California. The local economy survives on tourism since the "Alligator Mines" dried up (yes you read that right... It's probably the weirdest thing in the show... and the series featured a Lovecraft-esque writer trying to write his cosmic horror characters into a teen romance after a Stephanie Meyers parody mocks him for not writing what people want to read, so yeah, it's pretty weird). One of the big draws aside from the beach is the town's reputation as one of the most haunted locations in the nation... which causes the gang no end of grief because they are always solving the mysteries of the ghosts, thus ruining the possible new attraction. It also lends cover to the masked criminals (in one case the guy in the mask wasn't a criminal as none of his deception was an actual crime) as one more Ghost is expected and the local areas ready belief in non-rational explination as the town not only is motivated financially to go with the new ghost story, but also because there are some strange things about the town that would suggest there are some real ghosts and monsters. All the kids families now have a vested interest in the haunted tourism industry as they are either invested in the industry or the town's success and their own children are all embarrassing them by proving the ghosts aren't real.
The gang are also teenagers and this time their school life and family lives are fleshed out a bit more as well. Shaggy and Daphne retain their old money family roots, though this time Daphne's family also do not approve of her mystery solving hobbies and to a lesser degree, Fred. Their motive is that Daphne is the youngest of four or five sisters, all of whom are successful and all whom went to the same elite university their parents met at and clash with Daphne, who wants to be less identical. Shaggy's family has their traditional depictions of disapproval, but it's much more openly depicted. Velma's parents finally are given a job, here working class, as they are the curators the town's ghost museum, which Velma works as a tour guide after schools, though they quickly decide not to make her do this, as her parents leave out the fact that the ghosts were all proven hoaxes and Velma will rattle off the crimes of various ghosts (all from the original series plots) to any tour group.
The biggest difference is Fred, who is not from an Old Money family but rather his single father is the Mayor of Coolsville. As the duly elected representative of a town that hates both crime and revealing ghosts to be criminals, Fred's dad is also not in favor of his son's hobbies, which aren't helped by the fact that Fred in this continuity has some issues grasping social norms (he loves making traps... which is weird but... not to an alligator mine level. At least even his friends think he's a bit too obsessed). It's also quickly shown that Mr. Jones may not be the honest politician one would expect, given the town's tendency to look the other way whenever a ghost shows up and he has some knowledge of the overarching mystery presented in the series lore.
As far as continuity, it's a toss up, as the show all but says the original series ghosts all happened in Coolsville and were all solved by the gang. The Ghost Museum additionally features an exhibit dedicated to the character Flim-Flam (from the 70s-80s comedy era) who Daphne wonders what happened to him and Scrappy-Doo, which Daphne shrieks in terror when she see's his statue and when she tries to apologize and explain why she reacted like she did, Fred quickly reminds her that the entire gang swore they would never discuss the events surrounding Scrappy again. Given the show's blend of being way more darker and edgier than typical Scooby-Doo, but still paying a loving respect to the series invokes, it's hard to tell what previous were cannon, and as the series progressed, events would further throw this into doubt. However, given that the series was widely loved by long term fans, expect that the characterization of the families and some other popular characters would be reflective of this depiction (this particular depiction was homaged in an episode of Young Justice's season 3.).