“Arctic” is from the Greek arktos, “bear,” because the constellation Ursa Major, “the greater she-bear” (also known as the Big Dipper), is always visible in the northern polar sky.
“Antarctic,” therefore, literally means “opposite the bear.”
By force of pure serendipity, polar bears reside at the North Pole but not the South, making the Antarctic the land without bears in more ways that one.
The Proto-Indo-European root at play in the Greek word is *rkto-, which is also the root of many words for “bear” in Latin (ursus), Welsh, Armenian and more—but not in English.
The English “bear,” instead, derives the Proto-Germanic root *bero, literally meaning “the brown one” or “the brown animal.” (Hence it’s also related to the word “beaver.”)
It’s speculated, therefore, that the word “bear” is euphemistic, used in place of other, more specific (but now lost) words for bears that were derived from *rkto-. That is, bears were so frightening as game animals and predators that rather than speak their name—at rist of summoning them—people chose to call them “brown things” instead.
Note: This post has been updated with additional, clarifying information.
Useless Etymology 
This is a very interesting suggestion for how the Arctic and Antarctica were named, but it contradicts a much older origin claimed in this entry:
https://en.wikipedia.org/wiki/Antarctica
I like the idea of “Bears” and “No bears” better, but it may not be right?
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That entry does not contradict this post. It says “The name Antarctica is the romanised version of the Greek compound word ἀνταρκτική (antarktiké), feminine of ἀνταρκτικός (antarktikós),[10] meaning ‘opposite to the Arctic.'” The word “Arctic” is from the Greek arktos, or “bear,” after the constellation of that name.
The continent of Antarctica was certainly not named with physical bears in mind, but that is its literal meaning. I did not claim otherwise—I merely remarked on the serendipity of their presence.
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That’s fantastic! I love, love the parallel meaning, worthy of a crossword clue.
When did you first become aware of the etymology? Sadly, I was watching a netflix “historic” drama.
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Yes indeed, it’s all very clever, the multiple associations.
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