I know, it's an English site, not Swedish, but maybe someone will know: The Swedish word for barrel is tunnan whereas the word for thin is tunn. Does that suggest that the Swedes have skinny barrels, or are they from different roots?

They're not related. Swedish "tunnan" is related to English "tun" ("A large cask or barrel") and Swedish "tunn" is related to English "thin".

Just curious, Geoff - how did the question come up? You have the best questions!