Is there such a thing as a permanent academic assistantship position? It's common for graduate students and postdocs to fulfill various "assistant" positions in academia. They may grade papers or conduct research under the direct supervision of a faculty member who is ultimately responsible for the work product. These are functions that require real academic skills, but not up to the level expected of a faculty member. The important thing about these positions is that they are intended to be temporary, a type of brief apprenticeship training on the road to professorship rather than an actual stable career that one could stay in for a few decades.

Is there such a thing in academia as a permanent, full-time assistantship, one not temporary or tied to student status? That is, is it reasonably possible for someone to do this as a career? Obviously, this sort of thing probably wouldn't pay as much as a full-time, regular faculty appointment, but that is neither here nor there for this question.

I'm not thinking about Adjunct Professors. Adjuncts are usually temporary, part-time employees that actually hold faculty-level responsibility. I'm talking about a situation where a person might hold a full-time, indefinite appointment grading student papers, finding citations, or checking petri dishes for bacterial growth under the supervision of faculty, but wouldn't do (and isn't in training to do) their own research or teach unsupervised.

I also recognize that academic departments often have general office positions such as secretaries, human resources professionals, IT help desk technicians, etc., but these sort of roles do not typically involve the use of advanced academic skills.

There are various reasons a person could want to do this. They might want to work in academia but lack the talent to do truly original research, or they might actually have it but realize part-way through their PhD that they hate doing original research but truly enjoy grading papers or holding tutoring sessions. They might like going out into the field and applying best practices in digging up soil samples but enjoy the fact that their career is not in jeopardy if the research hits a dead-end (rather, the faculty that decided to dig there faces the setback).

As an example, someone might say, "Ok, I'm wrapping up my MS in Chemistry and I don't want to go for a PhD, my thesis was hard enough! I'd really like to spend the next ten years grading freshman homework, TA'ing the Organic Chemistry I lab, and designing data-gathering protocols for others' research projects. Those were the parts of the past two years that I truly enjoyed."
Yes, this absolutely exists in my field. Such people are typically called "Research Scientists" or "Research Associates." At my last university, a PhD was only a requirement for Research Scientist III. The level of independence depended on your own experience, and I suppose your PI's desires. That is, I would expect each one to have their own pieces of a project, and be on a few papers a year.

I have not heard of one having teaching responsibilities (even just grading) without holding a PhD, since those are often used for grad student funding. I do know of a few people who had lecturer responsibilities without holding a faculty position technically.