OK, so I have a 1 1/8" threadless headset on a mountain bike. It uses cartridge bearings, but they've got really rough. Adjusting the headset doesn't help, so I figure the bearings must be worn out. Is there some sort of standard for the cartridge bearings used in headsets?

Thanks

Not really.
What brand and model is the headset?

Either the headset will tell you what bearings it use or you can pull them and measure the bearing seat. There are various integrated "standards" for bearings. This page gives a brief overview. Alternatively bring those bearings into a shop if you are unsure.

https://www.parktool.com/repair/readhowto.asp?id=68

Depending on the type of bearing, you might be able to pick the seals off, clean the bearings and apply fresh grease. Just sayin'.