Unlike a lot of English words, it doesn't come to us from a foreign language. The root of this word is the Old English caru, meaning sorrow. The word is related to "care."
Chary means "discreetly cautious," vigilant, or slow to accept or give something (as compliments or praise.) It could be used as a synonym for prudent, careful, or fastidious.
If you are a person who gets the jitters from drinking too much coffee, you might be chary of drinking several cups of Venti Caramel Macchiato in a single afternoon.
Caution sign by Mike Licht/Flickr CC BY 2.0