Seeing a comet with the naked eye with 肉眼

Kanji Explanation: Seeing and insight with 眼

Before class, Asuka Jr. mentions a Graham Comet, which passes near the Earth once every fifty years.

涼子(りょうこ):: 「ステキ！！　それが　こんどのクリスマス・イヴにくるの？」; “Awesome!! Is it coming this Christmas Eve?”
アスカ Jr.:: 「どーせ肉眼(にくがん)じゃ見(み)えねーよ。天文台(てんもんだい)にでもいかないとな」; “You won't be able to see it with the naked eye anyway. You'd have to go to an observatory or something.”; Literal: “Anyway, you can't see it with the naked eye. You'd have to go to an observatory or something.”

Key Points

1肉眼(にくがん) = “the naked eye”

In this scene, 肉眼(にくがん) means seeing something with your eyes alone, without a telescope or other equipment.

2肉眼(にくがん)じゃ見(み)えねーよ = casual “you can’t see it with the naked eye”

じゃ here is the casual spoken form of では.
見(み)えねー is a rough casual contraction of 見(み)えない.
The ending よ adds an assertive “I’m telling you” tone, fitting Asuka Jr.’s matter-of-fact attitude.

3どーせ = “anyway / in any case”

どーせ is a casual spelling of どうせ.
Here it carries a dismissive nuance: “you’re not going to be able to see it anyway”.

4天文台(てんもんだい)にでもいかないとな = “you’d have to go to an observatory or something”

にでも softens the suggestion to something like “to an observatory or somewhere like that”.
いかないとな is casual for いかないと(いけない)な, expressing necessity: “you’d have to go”.