I have in mind units that people make up for the purposes of their own field, but are unlikely to be used (or even known) elsewhere. No, "leagues" and "short tons" don't count as weird.
Some on my list include:
- sverdrup (Sv), a measure of ocean currents used by physical oceanographers: 1 Sv = 106 m3/s (= 264 million gallons per second). The amount of water going over Niagara Falls, 1834 m3/s, is 0.002 Sv. The total of all fresh water emptying into the oceans is about 1 Sv.
- MegaYuri (MYuri), a measure of tether strength used by the space elevator community: 1 MYuri = 1 GPa/(g/cm3) = 100,000 N/(g/cm). Strangely, this unit has the same units as velocity-squared, and the speed of sound in dry air (343 m/s), squared, is about 0.1 MYuri.
- Einstein: A mole of photons.
- The barn, a unit of area used in particle physics scattering experiments (as in, "big as the side of a barn"). 1 barn = 100 fm2 = 10-28 m2, and is about the cross-sectional area of a uranium atom. Related units are the outhouse (1 microbarn) and the shed (1 yoctobarn, or 10-52 m2).
- This gives the "inverse femtobarn" (fb-1) used by particle physicists, which just may be the worse, least intuitive unit in all of physics. It's a measure of particle collision events per femtobarn; one of them is a big deal. Fermilab only produced 10 fb-1 in the last decade, and one experiment at CERN just celebrated their first one.
- Jansky, used by radio astronomers and heard by everyone who watched the movie Contact. 1 Jy = 10-26 W/m2/Hz.
- Warp Factor (w), as in speed, as in Star Trek. The definition seems to have changed over the years; in the Original Series a ships speed v was related to its warp factor w by v = w3c where c is the speed of light. Later apparently it was changed to v = w10/3c, kinda, because for some reason ships couldn't go faster than w=10.
- Cochrane (C), a unit of subspace distortion, whatever that is. Named, of course, after the inventor of warp drive, Zefram Cochrane.