Their academic rank is left ambiguous, but their jobs certainly seem to be almost entirely research, rather than classroom teaching. This is not unrealistic: There are plenty of people, at all levels of academia, who don’t do any classroom teaching, or do only as much as they want to.
The lack of grad students is more problematic. All competent permanent faculty will have multiple grad students working under them, and even postdocs will have multiple grad students working alongside them in an inferior position. The only time when this isn’t the case is at schools without a graduate program (this is certainly not CalTech), and even then, you’ll have upper-level undergrads instead.
It’s not quite accurate to say that postdocs will always outnumber professors. Every professor would like multiple postdocs, but there just aren’t enough of them to go around, leaving many professors without any. Plus, even when you do get postdocs, you can’t always hold on to them: Postdocs aren’t expected to have any job loyalty, and will leave in a heartbeat if they get a permanent position elsewhere. Again, though, at Caltech specifically, you’d expect to see tons of them.
We covered a lot of this ground in a recent thread.
Purely research positions do exist and are common even at places well below CalTech. I’ve also known a lot of regular tenure track profs who don’t teach simply because they “buy out” of teaching via getting a lot of research money. I used to regularly get 1/3 to 1/2 buy out with little effort. Getting 2/3 buyout was quite common. Going to full buyout was less common but doable.
When you are getting research money at this scale, money for grad students is definitely part of the package. A small grant might fund two. Getting money for 4+ wouldn’t be a problem for these guys. So, yeah, where are they?
It does appear that the PhDs are in research positions but they competed for a tenure slot in a recent episode. This is called “the writers have no clue how a university works or they do but needed it for the plot.”
In the US, not necessarily: several of the professors in my US grad school would moan like scared virgins any time they had to teach a class for a term. Many of them taught one term per year.
There are also situations where the titular professor is officially in charge but all teaching, as well as most exam preparation and grading is done by the TAs; I’ve seen this in Spain and the US (I was one of those TAs) and heard about it happening in other countries as well.