“Arctic” comes 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,” then, means “opposite the bear.”
Accordingly (albeit serendipitously*), polar bears reside at the north pole but not the south, making the Antarctic the land without bears in more ways that one.
*Evidently my meaning was not clear, so I’ve made an addition here.
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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Yes indeed, it’s all very clever, the multiple associations.
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