Could it be a feasible project for me to recreate the NASA technology using my own hardware and methods?
Sure, but beware the costs.
1960's cameras where impressive beasts, made from components (camera tubes) that no one uses these days anymore. To top this up, NASA certainly didn't just buy a stock TV camera and put a sticker on it, they certainly put thought into weight reduction and operation in space and with astronaut fingers...
Or at least learn how it was done in detailed theory, electronically, physically and mathematically?
I'd recommend you start out by studying electrical engineering, which is the theory of electronics, the physics behind electronic devices and the math behind practical signals.
If you have the basics of linear electrical networks down, it's probably worth taking a while to study signal theory, after which you turn to understanding tube devices (something that is not commonly taught these days anymore – tubes aren't used for these kind of things anymore, but are typically specialty devices for high-power microwaves etc).
Your signal theory will nicely prepare you for both understanding what the electrical signals coming out of the camera tube itself are, as well as understanding the basics of RF and RF communications.
A bit of self-study on legacy analog TV standards will probably be easy after that.
Then, you would probably be able to understand the functionaly of each (ok, most) components in such a camera in a detailed way.