There once was a soft water system in our house. I don't see a hand shutoff valve anywhere, other than the one that is clearly leaking... Ideas?

Of course I'm home alone with three kids when this kind of stuff happens...

Until you get to a store? Gallon ziplock bag folded over a couple time then secure over open pipe with a zip tie.

White plug pulled out, no threads of any type.

Flex tape might work.

Jb weld sounds better. Just plug the [censored] out of the valve... Maybe that'll work.

Open a bottle of wine and cork it.

Five gallon bucket and dump every hr. until you can get to hardware store.

Might try getting a threaded (metal might work better than plastic) plug that would fit tight, coat it with teflon pipe dope and wrap with teflon tape and drive into the opening with a hammer. Just a stab in the dark at a temporary fix.

Remove insulation, check pipe type. I usually do not recommend shark bite but shark bite fittings could work.
I am assuming you will need to tie pipes together for water to house and remove old header.

Get a crescent wrench and make sure the valve really is closed. Sometimes those 1/4 turn valve handles wallow out. You may have to remove the handle. Anything you try to plug it with will eventually build pressure and leak.

Plug it with a ball of JB water weld putty then go to store to purchase replacement valve.

If you don’t have any JB water weld, make sure you have some available at home, on boat, and kayak at all times going forward.

Other temporary fix options:
garden hose over the spout. Secure with duct tape. Run garden hose outside or to the shower drain.
Or
Just let it drip into a bucket till you get back
Or
Closed cell foam cut into a cork shape, stick it in there, then duct tape.
Or
Double ziplock bag barely into hole. Zip tie. Fill the small part of the bag that is in the hole with a little bit of spray expanding foam. Then duct tape over everything.

Thar cut off handle next to the wall goes to what.

take plastic you removed to hardware store, find a pvc cap that fits, remove o ring or leave it, your option.

glue cap on plastic, use some jb to reinstall plastic to brass so it seals.

that will not make pressure, looks like a bad gasket type leak.

or put kids in car, light fire in garage and go to schlitterbahn while house burns down, when you rebuild you can eliminate the bs water system.