It's actually not that bad: A pi zero can do most of the tasks balanga is envisioning (sprinkler controller), and with wireless uses about 120mA at 5V (there are several web pages with that number). So over a 12-hour dark night, it needs about 1.5 Ah at 5V, or about 7.5 Wh. But any ancillary electronics will increase that, perhaps significantly.
But we can't just run it with a battery that has just that capacity, because (a) it would be mostly discharged every 24 hours, causing rapid wear out, and (b) if you ever get 3 days of bad weather (very rainy and overcast), your battery will actually go completely empty (deep discharge). I would oversize the battery by a large factor to (a) keep the stress on the battery low, and (b) allow for bad weather. A nice option would be a standard-size 12V 7Ah battery (about 2x4x5 inches, and not very expensive), which can provide roughly 100Wh. That is a very comfortable safely margin. Converting the 12V down to 5V is easy (very efficient DC-DC converters can be found many places), and charging systems and solar panels for 12V are very common (because of the automotive / motorcycle / industrial market).
And finally we need to be able to charge it. Say that we want to aim on average for 8 hours of sunshine per day. During the other 16 hours, the RPi uses about 0.6W, so during the 8 hours the panel has to deliver 3x that (1 + 16/8) to run both the RPi and recharge the batteries. That would make for roughly 1.8W. To account for the inefficiencies (the 12V -> 5V converter is only 90% efficient, the solar -> 12V battery charger is only 90% efficient, and the battery itself burns 10% of the charge in the form of heat), we'll round that up to 3W (I'm being generous). A solar panel that makes 12V at 3W is easy to find, and about 25x25cm in size. With battery and charger, the whole setup should cost less than $100.
An interesting question is: Sprinkler valves (solenoids) need 24V AC, *not* DC. Finding inverters that make 24V from a battery seems hard, and two minutes of google'ing didn't find any in small and inexpensive sizes. This part along might double the cost of the project.
