Background:
I was looking at the system a few weeks ago since I had noticed some water in the box occasionally a few months ago when they were running regularly. My investigation a few weeks ago found that the front sprinklers (different green, in-ground box than the backyard sprinklers) worked fine, but the rear/backyard sprinklers did not...just an occasional humming from what I believe to be the solenoids.

Today it was raining and the green box was full of water. The wires have water-tight wire nuts on them, but I don't know if one of them has failed. One zone turned on by itself and would not turn off. The control box is in the "off position". I tried the other zones in the backyard and magically they work again -- albeit at a much reduced water pressure because of the one zone that would not turn off.

Anyway, I ended up having to shut off the water to the sprinkler system to get it to stop. Any ideas what's going on here? Seems to me that it's electrical. Does power have to come from the control box for the sprinklers to turn on? Please help.

How many zones(valves) are you running from the controller?
Your problem is most likely electrical and even needs replacement of the controller. Since the system is of indeterminable age. I would do a replacement of the controller. Cost relates to the number of zones you have(around $50 to $125). A test would be to turn on and off the solenoids at the valves and see if they work properly(slightly twist counterclockwise till zone water comes on and then slightly turn clockwise till zone water goes off, do not over tighten). If there are no problems with solenoids, then it is the controller or wires to the controller . GL Aloha

My solenoids are all turned clockwise/off. Is that where they're supposed to be for the controller to activate them and turn on each zone?

Yes, but to make sure the mechanics of the solenoids are operating properly, complete the exercise I describe before and turn on and off each zone and observe that it actually comes and then goes off by turning the solenoid. Again do not over-tighten, turn clockwise by close only enough for the zone to go off. Let me know what happens. If you replace your controller you would need an 8 or 9 station controller. Aloha

Most home systems with multi-station systems have small rather inexpensive diaphrams. If the valve is several years old the parts are past their prime anyways. Often debris get in valves because of work on city water lines or because of dirty water BUT often if the debris get in the valve to cause them to stay open they have somewhat or completely damaged the diaphram anyways.

One thing to be very careful of is to not let dirt or rocks get in the valve while you have it apart. Almost certainly your going to need to keep the valve apart while you get the diaphram so either reassemble the valve missing the diaphram or cover it so nothing can get inside. Several years ago I left a valve open that was clear of any possible dirt. Over night a frog got inside it and up the pipe enough that I didnt see him.....yea that was a mess.

Good luck,
Tom