I think it's safe to say, based on the character's race, gender, and where he works, 50 years is a fairly representative career lifespan for a teacher in his position. Context clues in the episode lead us to believe he's in his late 70s or 80s in 1962 when the episode was filmed, which would mean he started in his mid-20s, which would be reasonable. I wasn't able to get a sense of when the yearbooks he's looking at were published, but such "year books" (as they were known then) were fairly common at colleges, military academies, private schools, and some public high schools. They emerged in the early 1900s so it's reasonable he'd have a collection of such books going back to the beginning of his career. There are, though, a few things that may have interrupted his career, most notably military service. He talks about the death of some of his students during World War II and I'll defer to those who know the history of military recruitment, drafting, and deferment regarding why he didn't serve in either war. Finally, I don't have access to extended demographics on private school teachers so it's possible he's an outlier but it strikes me as a routine thing that a young man would join the faculty of a private New England boy's school and never change jobs.

All of that said, the reason he was able to have a career that spanned 50 years in the 20th century was his race and gender. First, and granted I didn't watch that closely, but I think they wanted us to understand he'd never been married or had children and teaching was his whole life. Had he gotten married, though, there would have been zero expectation he quit his teaching job. Likely nothing would have changed for him, regardless of this teaching position, seniority, school, or geographical location in the country unless he wanted something to change. That same could not be said for women teachers.

What happened after a woman teacher got married was highly contextually dependant and linked to the local supply and demands of teachers, the social norms of the community, the degree of union solidarity in the district, and even the woman's individual choices. I get into some of that history in a response to a question about a so-called teacher contract that often makes the rounds on social media.

The marriage issue is accurate but they've put the cart before the horse. A woman would typically and voluntarily leave teaching if she got married as a consequence of institutional sexism was that people thought a woman couldn't be a wife and a teacher at the same time. However, there is an entire history of teachers keeping marriages, gay and lesbian partnerships, and pregnancies secret but they're geographically and time specific. Additionally, marriage bans ebbed and flowed with population booms and the availability of men teachers. Chicago schools cracked down on married woman teachers during the depression and eased up during and after the war but part of that was due to the personalities of the men in charge. You can see the changing social norms around married teachers play out in this 1935 episode of "Our Gang." (The big reveal comes at 13:00. Be sure to watch through for him to announce he's going to "let" her keep teaching.)

At the same time, Professor Ellis Fowler's career wouldn't have been negatively impacted by his race. If we assume the show is set in 1962, that's less than a decade out from Brown v. Board. Granted, private schools like the one he's at experienced the ruling differently than public schools, but had Fowler been at a public school, the ruling would not have impacted him in any meaningful way. However, Black teachers of all genders, especially those who taught at schools with all Black student bodies, found their career trajectory dramatically changed by the ruling. In many cases where schools were segregated de jure (meaning by legal and school policy), Black schools were simply shut down, sending the Black staff and students to white schools. I get more into the impact of Brown on Black teachers here. From that answer:

Numbers vary, but it's estimated up to 38,000 Black teachers lost their jobs as a direct result of Brown v. Board. (Here and here) while very few, if any, white teachers did. The impact on an individual level was dramatic and profound. In one case, a highly educated Black physics teacher who was also the football captain at the closed Black school was told his only option was to work as an aide and an assistant to the football captain at the white school. Women with multiple degrees from HBCUs who taught literature and Latin were given the choice of Kindergarten or unemployment. While modern day teachers are more comfortable with the idea of co-teaching - multiple adults in one room - many white teachers refused to share space with incoming Black teachers or otherwise froze out the new faculty. Black teachers were often forced to choose between working in a school that did not want them where they might be able to be a resource for Black children or protecting their own personal dignity and self-worth.

Finally, it is possible there were women teachers who taught at a private girls school for 50 years while getting married and raising a family without encountering pressure to resign. There are several women who come to mind who I know taught at public schools for 50 or more years - and only a few of them that I'm thinking of married. To return to your question, you asked about what was representative and the teacher in this episode does not represent the typical American teacher in the 20th century. The "average" teacher was - and remains - a white woman. In the 1960s, she was younger than the average teacher is now as it was common for a woman to leave upon getting married or having children.

A momentary opine: It would have been possible for Sterling to place the exact same story at a private boarding school with a mostly Black faculty and student body. Although most closed in the mid-20th century following desegregation, there were numerous private Black boarding schools that a Black male teacher could have worked at during the same time frame as Professor Fowler. As of the time the show was set, about 10 or so of them were still open. Like the Rock Spring School for Boys, the schools focused on a classical curriculum, literature, poetry, and coaching and encouraging young men down a particular path towards adulthood.