The weirdest thing I own?
Probably a large wall tapestry from the early 1970s shows 2 tigers in a jungle. My dad was in the Air Force, and bought it during a trip to Libya. It's sort of like this:

It's packed away in storage quasi-permanently. It's too kitschy and weird to get rid of, but it hung on the wall in my grandparents' house for about 20 years... and they were very heavy smokers. It smells absolutely horrendous. I tried airing it out for an entire summer, and everything possible to get the cigarette smell out of it, but no luck.