Reasonable CPU power usage for solar panel powered system with battery
As you say, tough crowd, especially as you've done a bit of research into your actual power consumption.

Your device consumes 5V * 1/4A = 1.25W. Lots of solar equipment is available for use with 12V lead-acid batteries, as used in caravans, so let's assume that. Let's assume that you drive the device through a buck converter (switchmode power supply), so the current draw at 12V is about 100mA. Let's round that up to 125mA for headroom. You should allow at least 48h of operation (through dark days in winter), so you need at least 6Ah of usable capacity from the battery. Choose deep cycle gel batteries, although for this level of discharge a car battery will be an OK substitute if you have one lying around.

Now you need to size the charger. Allow twice as much charge into the battery as you expect out, for charging over an 8 hour period. 12 / 8 = 1.5A charge current. Note that this is a lot more than the drain current! That's 18W. So e.g. this is a little undersized, but the right kind of thing to be looking for. You could buy a panel and charger; whole thing should be under 1m^2.

Don't forget to align the panel optimally and weatherproof everything.