Your sprinkler controller is likely going to "want" to have 24-ish volts of AC to run the valves anyway. (There are DC valve solenoids, but 24VAC is more common.)
Given that, I'd run a 120VAC to 24VAC transformer someplace dry, then carry the 24VAC across the yard, underground, whatever you need to do, and near the project box turn that into whatever your project needs. That could be another transformer, bridge rectifier, and linear supply, or a switching supply, depending on your interest in selecting or building the various options.
I would not run 5VDC over a long run (due to voltage drop concerns). This is much less a problem when starting with 24V than 5V, of course.
As for having high and low voltage in the same box, it's done all the time and can be done safely, but if you can manage for your box to take only 24VAC, almost every safety concern you have goes out the window, plus it happens to be what you're going to want to switch with relays to send out to each zone box anyway.