First Sentence should either be "I'm going to Antarctica for 6 months" Or "I'm going the antarctic for six months." Antarctica is a specific place. The antarctic is defining a general region and regions like that are usualy prefixed with the word "the." For example you cant say "I am going to Americas" It would have to be read "I am going to the Americas" conversely you can say "I am going to America" but it sounds odd if you say "I am going to the America" Now the question you wanted answered. The correct way to say it "You're going there for quite a long time." The word quite is an adverb adding meaning to the word "long." The sentence should make sense if you take the word quite out. Without it the sentence reads "You're going there long." which doesn't make sense because there is nothing attached to the word long for it to describe. If you said this to someone they would obviously understand what you were trying to say. It is obvious you are describing "a long time." But in strict grammar terms what you have written is incorrect.

*You're going to be there quite long* would be a better sentence!