Going to label those with "millionaire" families in italics.
Hamilton's story is well publicised. With Lewis' dad who was working in IT as his main backer early on, the duo pushed through plenty of obstacles early in Lewis' career. Anthony took on additional jobs and put everything he could on the effort, until Ron Dennis finally snapped his son up around 1998.
Bottas is the son of a small cleaning company owner who estimates he spent some 200k euros of his own money on his son's karting career. His mom is an undertaker and Valtteri himself is a qualified car mechanic in Finland, so he probably would be one if not for F1. Upon entering junior formulae Bottas got sponsorship from local companies Wihuri, Kemppi, and in 2008 got signed up by a manager team involving Didier Coton, Toto Wolff, and Mika Hakkinen.
Vettel doesn't have a wealthy background either as his dad owned a small carpentry business - but luckily for Sebastian, he was also into motorsport and their home was located nearby the local karting track, so with his contacts he could fund Seb's karting from a very young age. The owner of the said track started backing Vettel at a young age, and Red Bull Junior Team signed him around the year 2000, so presumably the Vettel family didn't have to invest quite as much money as most others.
Raikkonen comes from a family of petrol heads who didn't even have an inside lavatory and he, too, would most likely be a car mechanic now, if not for a lucky break with the Robertsons spotting him on his first junior formula season in 1998. Only Kimi's road builder dad working some additional jobs and taking on loans had been keeping Kimi's karting career going until that.
Verstappen is obviously the son of Jos Verstappen, a former F1 driver in the late nineties and early noughties. With his money and contacts (such as the Jumbo supermarkets), Jos was able to fund Max from karting, all the way to F3 in 2014 where Red Bull snapped him up.
As noted above Ricciardo had fairly wealthy parents, his dad was the chairman of GR Engineering so they were able to fund young Daniel through the Australian junior formulae and a season in Italian Formula Renault in 2007 - after the season Red Bull snapped him up.
There's not that much online on Hulkenberg's father other than that he owned a shipping company and probably funded his career throughout his karting years. When he stepped into cars around 2005, Willi Weber (Schumi's manager) took him up and managed him all the way to Williams with some backing from Dekra.
Sainz jr is the son of Carlos Sainz sr, a double World Rally Champion. Probably his dad's wealth and existing connections with Red Bull didn't hurt him when Carlito stepped up from karting in 2010 and instantly got signed up with their junior team.
Perez comes from a wealthy family, saying he would probably have become a lawyer if not for motor racing. His dad used to be a racing driver who had some existing connections with Carlos Slim (the owner of Telmex and one of the richest men in the world) who funded Checo as well as many other Latin American drivers from the early 2000's on.
Ocon does not, his dad owned a small garage who invested everything in young Esteban's karting career, eventually ending up selling their house and living off a caravan. Their money finally ran out around 2010, still in karting - but just then, Estebans career was salvaged by Eric Boullier who promptly signed him for Gravity Management.
Magnussen is the son of Jan Magnussen, a former F1 driver slash current sportscar driver. Not the wealthiest as far as racing driver daddies in the grid go - but still, the name mustn't have hurt when collecting sponsors. Most prominent has been the Danish company Jack & Jones who supported Kevin's career into car racing and eventually into McLaren Young Driver Programme in 2010 too.
As also noted above, Grosjean comes from a family with plenty of skiing heritage. His dad is a Swiss lawyer and mom a relatively famous French painter. Romain himself wasn't always going to be a racing driver, he only started at 14 and spent some of his karting years part time working at a Geneva bank. His family presumably funded him through his karting times and early junior formulae, until 2006 when he was snapped up by Renault Sport Academy. As an interesting tidbit, after his "first" career in F1 went bust, the disenchanted Grosjean promptly decided to quit racing and chase another dream: to become a chef. However, the cooking school didn't let Romain enroll because he was too old, and thus forced him to try and salvage his racing career instead!
Alonso's dad was a mechanic in an explosives factory slash amateur kart driver. His parents were thus willing to support Fernando throughout his karting years, until 1996 when Adrian Campos spotted him and offered him a ride in his junior formulae. A few years later Briatore would spot him and took him to Minardi in F1.
Vandoorne comes from a family of no racing heritage. His father was an architect who just designed the restaurant at the local karting track - unfortunately for him though, his then-6-year-old son got enamored by the karts there. Hence the family went through a decade of financial hardships to gather local funding for Stoffel, and it was only prize money and scholarships gained from winning various karting & junior formula competitions that kept his career afloat until 2013, when McLaren Junior Team finally snapped him up.
Leclerc's late father was also F3 driver in the eighties and nineties, and they lived in bloody Monaco to boot, so I will italics him - even if he claims his dad was not *that* rich and ran out of money in 2010. But the Leclercs' close family friend Jules Bianchi then spoke to his manager Nicolas Todt, who subsequently signed Leclerc up, and this connection later helped him sign with the Ferrari Driver Academy too.
Ericsson easily gets the rich pay-driver tag because of his large sponsorship. However his dad was a house painter with no motorsport background whatsoever, so the Swede was anything but rich. His first taste of motorsports came at age 9, when Marcus simply turned up at the local rental go-kart track and nearly broke the lap record off the bat. As the small town of Kumla also just happened to be Sweden's main motorsport town, the kart track owner-racing driver Fredrik Ekblom was there to instantly recognize his talent and start funding him, and later on Kenny Brack and Eje Elgh to help out some more. With their extensive business connectionsin Sweden, Marcus was later able to land substantial backing from Tetra Pak, H&M and Olsbergs, which, combined, eventually brought him up to F1.
Gasly comes from a family of racing enthusiasts, and his father did race in various disciplines. Pierre claims they were not very wealthy though, and that he managed to get all the way up to F1 "without any manager and without paying for my seat". However, as I cannot even spot any recurring sponsors on his junior formula cars, I suppose at least some of the early racing must have been funded by his own family, until he landed Red Bull backing in 2014.
Hartley's dad was a local racing driver who even finished P2 in the junior formula New Zealand GP in 1995. The Hartleys were major petrolheads and still have their family business that makes high performance engines. They were able to fund Brendon's karting career until age 14, when some local Kiwis set up a shareholding platform to support Brendon's career, but even that wasn't going to be enough to step into the big time. So, desperate for cash and opportunity, they finally reached out to Helmut Marko - and got lucky. And as is well known, Hartley didn't initially make his breakthrough in European junior formulae, but eventually got his F1 chance through a "detour" of years of fine performances in various sportscars. (Today I also learned that Hartley's #28 was a tribute to Jean Alesi. Who knew.)
Stroll's dad Lawrence is worth 2.7 billion USD, and funded his junior formula career throughout, even buying stakes of the Formula One teams Lance was going to join.
Sirotkin's father Sergey sr is the head of the National Aviation Industries in Moscow, and Sergey jr himself has graduated from university with a degree from race car engineering. Daddy's company has largely funded Sergey junior's early career, and SMP Bank, among others, stepped up when it came to funding his Williams seat in 2018.
Russell comes from relative wealth as his father was involved in "lucrative beans and peas business" until selling up (apparently - I regret to admit my only source here is Daily Mail, so ...). So they were able to fund George's career for quite a while. Around 2014 Russell contacted Toto Wolff and signed with some kind of junior programme he had.
Kubica's father was involved in business, and a "fan of racing", but he had no racing background and thus no proper connections. Kubica could get his karting license in 1994, but as the karting infrastructure in Poland was poor, at the age of 13 his dad loaned some money and the pair of them went to Italy without any sponsors. Having done this, Robert landed a deal from CRG, a kart manufacturer, to test karts for Nico Rosberg. Kubica went up to formula racing largely with the support of Daniele Morelli and Renault Sport Academy, until he got snapped up by BMW. Later in his career, after the rallycar accident, he would gain backing from the local oil company Lotos.
Norris definitely has a rich dad, a retired pensions manager, who funded his career all the way until McLaren signed Lando up in 2017.
Kvyat has a rich father too, he worked at a Russian oil company and later became an influental politician from the Republic of Bashkortostan. So they could afford to fund young Daniil in karting and support his career by moving to Rome when Kvyat was 13. In 2009, while still karting, Kvyat was snapped up by Red Bull who have funded the rest of his career so far.
I can find little of Giovinazzi's dad. Antonio started karting at a young race and karted for 10 years, so presumably they were quite wealthy - though I'm happy to get corrected if someone knows. In 2015, he raced Sean Gelael, whose father is very wealthy of KFC, and who also took a liking in Antonio so he decided to fund Giovinazzi's career as well.
Albon has a father Nigel who was a part time Porsche business owner, once raced in the BTCC - and a Thai mom who is currently jailed for a 7.5 million pound car fraud(!). So I presume they were wealthy enough to fund Alex's kart career. He was signed up by Red Bull junior team in 2012, then dropped for multiple years and picked up by the Gravity team for a while. And this year, obviously, got his break in F1 by Helmut having to crawl back to him.
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In conclusion: currently we've got a rough 50/50 split between the super privileged, and those only privileged enough to have their family "only" able to fund their karting career and then get a lucky break with something. But still, I wouldn't call many of them even middle class.
Now this was a fun end to my Christmas Day having eaten way too well, maybe I can use some of my time tomorrow for a review of the past World Champions' background too...
) - I do find it interesting to see where the drivers came from. Reading stories about the "normal Western kids" who made it through plenty of hard work, great sacrifices and some strokes of luck, also steals a couple of the traditional excuses from the rest us lower middle class Westerners when it comes to the question "why am I not an F1 driver". 
