The expression "as X as a Y" is a common idiomatic expression to compare two things, often figuratively.
He is as tall as a tree!
One option is to compare this to a figurative maximum, to mean extremely something:
She is as smart as anything (she is extremely smart)
He's about as rich as a person can get. (he is extremely rich)
Adding "just about" to this phrase emphasizes the comparison
She is just about as smart as a person can get.
In the context of your example, "just about as brainwashed as a person can get" means "extremely brainwashed". The father believes that being a teaching nurse, and subject to the control of both the "Medical Establishment" and "the Government", makes Mary extremely brainwashed.
Note there is not enough context in this one sentence for us to know how the narrator feels about the father's opinion -- but we can assume more information will follow.