I have a standard old-timey one-handle Moen "mixer" (no model name/number/style) in the shower that has a slow leak up through the output pipe, so the shower head is dripping slowly throughout the night.

Is there a repair kit for this? Home Depot and OSH say no...
Seems like there should be something!

Where would the shutoff be - the shutoff for the whole house? Or would there be a shutoff for just the shower somewhere?

Sinks and toilets almost always have shut offs under the unit but not for showers. You likely do need to shut down your main and open a faucet to blow off pressue. Remove the handle and the escutcheon plate. That old Moen probably has a ring with two flat spots opposite each other, put a spanner onto the flat spots and unscrew the retaining ring. There should be a cartridge inside that can be pulled and replaced. Don't always believe what HD tells you.

Is there a repair kit for this? Home Depot and OSH say no...
Seems like there should be something!
This is why big box salesmen aren't outdoing plumbing. I'm amazed that they didn't sell you a #1200 or 1201 Moen replacement cartridge,( see image).
Remove cap (kitchen) or insert button (bathroom) to expose screw.
Remove the screw and pull off handle assembly.
Turn off shut off valves underneath the sink.
Remove retainer clip (copper horseshoe shaped clip) and pull out cartridge with channel locks, pliers, or cartridge puller. Lube new cartridge with silicone jel and
Insert the cartridge with hole on the stem facing the 6 o'clock position, or down in a tub/shower.
Reinsert retainer clip and screw in handle assembly, followed by the cap or insert button. Turn water back on under sink. My company uses Moen valves on all their new construction. Moen makes a "cartridge puller" That will get those stubborn cartridges out of the valve body if you have a problem.

Where would the shutoff be - the shutoff for the whole house? Or would there be a shutoff for just the shower somewhere?
Remove the handle and face plate. There should be screwdriver stops on either side of the valve body where the supplies connect to the valve. If not shut the house off at the house shut off located, as a rule, where the water service enters the house. Good luck, Tom

This was exactly what I needed!