“Happy Jack” by the Who might fit this category.
Or it might not. Who the hell can tell?
A lot of the Decemberists songs have unusual topics, but one that grabs me is “When the War Came” about the 900 day seige of Stalingrad. It’s apparently about a horticultural curator who’s swearing to preserve heirloom seeds at the risk of his own starvation.
Warren Zevon’s “Roland the Headless Thompson Gunner,” is about, well - a Danish mercenary who fights for the Congolese in Africa, gets his head blown off, and wreaks vengeance as a headless ghost. He also has an influence on Patty Hearst.
I’d say THAT qualifies! 
Sparks “Talent is an Asset” - about Albert Einstein’s parents attempts to keep him from distractions while he studies.
“'39” by Queen is about time dilation.
Safety Dance! I think it is an anti-nuclear power song, but the meaning gets lost in the silliness of the video
Rock me Amadeus It’s about Mozart
Down Under About Australia, but I am sure there is a double entendre there
Pretty much the entire 80s 
Although not explicitly stated in the lyrics, Ke$ha’s “Supernatural” is about having sex with a ghost.
Just for starters, the Beatles have songs about a submarine, paying taxes, and a guy who wants to write paperback books.
Well, a lot of hugely popular songs are about strange topics, if we actually think about them.
We’ve all heard “Come Sail Away” by Styx a million times, but we forget what an odd subject it has. It’s about a guy going through a midlife crisis who heads into space with aliens on a UFO.
“Another One Bites the Dust” is about a drive-by shooting, which was unusual when it came out. I imagine gangsta rap deals with it somewhat regularly, but would enjoy being contradicted on that.
Just about everything in Frank Zappa’s catalog.
Donovan’s “Mellow Yellow” is supposedly about a dildo, but was widely thought to be about smoking banana peels.
Pink Floyd’s first British single, “Arnold Layne”, is about a guy who steals women’s underwear.
The Who’s “Pictures of Lily” is about a kid who falls in love with a pin-up girl.
And a few self-evident ones…
Peter and Gordon’s “Lady Godiva”
The Royal Guardsmen’s “Snoopy vs. the Red Baron”
I’m sure I could come up with tons of these given time (hell, Rush’s catalog is full of them!) but the one that comes immediately to mind is “Down at the End of Your Road” by Jethro Tull, which is about a crooked real estate agent who does things like leave dog crap in mailboxes and put broken bottles in flowerbeds to encourage people to sell their houses.
Floyd the Barber by Nirvana. Sexual assault and death at the hands of the denizens of Mayberry from the Andy Griffith Show.
Braineaters - The Misfits Complaining that they always have to eat Brains for Dinner, Lunch, Breakfast Brunch and would really like some guts.
Scatterbrain - Down with the Ship (Slight Return) Ship is sinking - he wants to abandon ship. Or the band wanted to play as many riffs from other songs in one session. (From Star Spangled Banner to Woody Woodpecker)
Any song from GWAR - but special mention to SICK OF YOU where they are just sick of you…or SLAUGHTERAMA which is a game show where the contestant is killed if they don’t answer correctly.
Jim White’s “The Wound that Never Heals” is about a female serial killer.
Jill Sobule’s “Sonny Liston” is a song about the lack of permanence of physical memories.
Cheryl Wheeler “Estate Sale” about digging through dead people’s stuff.
Dar Williams “When I Was a Boy” about growing up as a tomboy and trying to reconcile that with societal expectations when the writer is an adult.
Evelyn Evelyn “Evelyn Evelyn” about growing up as conjoined twins.
Hart Rouge “Marie Caissie” about a couple who get in an argument at a party and the wife walks home only to freeze to death on the side of the road on her way home.
John Southworth “Pumpkins” about a small ex-mining town that is now attempting to maintain its former financial glory (but failing) by switching to pumpkin farming.
Imani Coppola “State of the Art” about being too out of step with former humanistic qualities as they’ve been overcome by technology.
Momus “Three Wars” about the arc of the life of a person that follows the 20th century wars.
Sage Francis “Makeshift Patriot” a drippingly cynical indictment of how media handled 9/11 coverage.
Sally Fingerett “Home is Where the Heart Is” a song about explaining to a child how long transcends sexual orientation.
I wanted to be the first to mention Sparks; they are the kings of songs with oddball subject matter. The first one that came to my mind was “Up Here in Heaven,” about two lovers who made a suicide pact, only the girl chickened out (“Up here in heaven without you/It is hell knowing that your health/Will keep you out of here/For many many years.”).
Serge Gainsbourg’s Les Sucettes, performed by France Gall when she was 18, might be one of the more notorious ones.