Learn German: Das ist absolut unmöglich. - That is absolutely impossible.

Yes, both mean almost exactly the same thing. However, when using a word that already contains a negative prefix like unmöglich (impossible), German speakers usually amplify it with words like absolut (absolutely) or völlig (completely) rather than standard negation amplifiers like überhaupt nicht (not at all).

What happens if I try to amplify this sentence with gar nicht, saying Das ist gar nicht unmöglich?

You would completely flip the meaning! Because unmöglich is already negative, adding gar nicht (not at all) creates a double negative. Das ist gar nicht unmöglich means "That is not impossible at all," which actually means it is entirely possible.

Could I use keineswegs in this sentence to sound more advanced?

You could say Das ist keineswegs möglich to mean "That is by no means possible." But if you combine it with the negative adjective and say Das ist keineswegs unmöglich, you are again creating a double negative meaning "It is by no means impossible."

Are there other adjectives that behave like unmöglich here?