When does the sci-fi book and TV series The Expanse take place?
Probably around 2350.
The timeframe for the series is, presumably intentionally, left vague.
The world of The Expanse started as a tabletop RPG run by Ty Franck (one of the authors writing the book series behind the pseudonym James S. A. Corey). This RPG was called "2350", and was presumably set in that year. (An anonymous Wikia user claiming to be Ty Franck heavily implied that this is the case.)
The TV series was first pitched as “[a] thriller set two hundred years in the future”. The same anonymous user suggested this was a marketing tagline and not canon, which seems likely if we look at the evidence; we can at least work out a timeline from the novels, by going backwards from what we know.
In developed nations the average familial generation length is in the high 20s and has even reached 30 years in some nations […] Conversely, generation length has changed little and remains in the low 20s in less developed nations
So Mars was colonized some 60 to 90 years prior to the beginning of "Drive". (depending on whether or not we think life expectancy has risen or fallen on The Expanse's technologically advanced, albeit polluted and criminal, Earth). Ten years later, Solomon invents the Epstein drive.
Leviathan Wakes takes place around 150 years after the invention of the drive, according to that novel's opening paragraphs. In the sixth episode of the second season of the TV show, "Paradigm Shift", we learn that that episode takes place 137 years after Epstein's first (and last) voyage with his drive.
So that'll put it around 200–250 years after Mars is colonized. We also learn in Leviathan Wakes that it's already around 100 years since the Canterbury helped colonize the moons of Saturn, so we know humans established colonies outside the Asteroid Belt 120–150 years after they colonized Mars.
So if we take an educated guess at how far in the future a colony on Mars is, we can pretty much estimate the time frame of the series from there. It also depends on whether or not we believe NASA will get more funding some day, or if we have to rely on private ventures such as Mars One or SpaceX. Of course, private companies versus governments is actually a theme in the books, but I'll have to reread the books to get an impression of whether or not Mars was privately or publicly settled. The fact that the UN has Mars under its heel in "Drive" makes me lean towards publicly, though.
Mars One claims that a Mars colony established in 2032 is feasible. Their roadmap isn't drastically different from that of SpaceX. NASA, however, plans manned missions (not colonies) to Mars in 2035.
Now, the authors of The Expanse and The Martian recently decided that their novels take place in the same universe. We know that The Martian takes place in 2035, which fits with NASA’s real world estimate.
We also know that the seas have risen on Earth due to climate change and global warming. We don’t know how much exactly but someone (me) approximated it to between 9 and 13 meters in this answer.
Although this is a well known matter of contention (and we don’t know what sources the authors used), according to some sources, the sea could rise 50 feet by 2500. That’s about 10 feet per century, luckily, and 9–13 meters is 29–42 feet. Following that specific projection, that puts the book series in the 2290–2420 range. Again, this is just one study – most of the other studies I found actually had similar numbers, but didn’t extend beyond the 2100s in their projections – but it fits.
So, all in all, the series would have to take place at least 240–270 years in "our" future – likely more. Ty Franck putting it at 2350 seems plausible, and fits with projected real-world scenarios like Mars exploration/colonization and global warming.
If you use The Martian's date of 2035, Mars must have been a UN colony for at LEAST 150 years before the Epstein drive, maybe more since 2035 + 150yrs + 150yrs (years since Epstein) its still only 2335.
Or who knows, maybe it took longer to colonize Mars in earnest after Mark Watney's trip.
We can also work it backward assuming Leviathan Wakes takes place in 2350. Then 150 years earlier, Epstein is invented in 2200. 90 years before that 2110's is when Mars was being fully colonized most likely after the invention of regular fusion drives (Pre-Epstein).
So perhaps, Mars was explored occasionally or in a small scale for 70 years (ie with our current technology)before the invention of Fusion drives thus prompting a rush from the nations of Earth to "pour their entire economies into space travel" and thus the true beginning of the colonizing of Mars and Luna around the 2100's. with the Belt being colonized heavily during the 2200's.
The TV series looks to be around 2215. At the very beginning it states that it is the 23rd century.
In the 23rd Century humans have colonized the Solar System
Then when Miller is looking at the possible courses for the Scopuli most of the planets in the solar system are shown and their positions match around 8/15/2215. The first image shows the positions of most of the planets and the second is a scene from "Back to the Butcher" with a clearer image of the inner planets. The last image is from www.fourmilab.ch showing the position of the planet on 8/15/2215.