48Vdc is a lot for a typical consumer WiFi device, but ok. The steps to calculating your system should go like this:
First Determine:
System Voltage = 48V
System Current = ??
System Power Consumption = 48 * I
Number of Hours of Operation Per Day = ??
For now I will Assume:
I = 1A
H = 24 Hours
From here we can determine the minimum battery size:
24 hours * 1A = 24Ah
Next we need to know something about the amount of full sun in the place the device will be located on the Winter Equinox. I will assume this is 4 hours.
You will need to charge the batteries back to at least 24Ah each day. I will assume an 80% charging efficiency, so:
(24Ah / 0.80) * 48V = 1440W / 4hours = 360W
You would need a minimum panel size of 360W @ 48V.
Next you need to evaluate how critical the device is to determine and calculate how long you might want the device to withstand adverse weather conditions. Let's say the device is highly critical, and you want to ensure there are NO interruptions for up to 7 consecutive days of 100% cloud cover.
So, our minimum battery size would be 24Ah * 7days = 168Ah
To charge our 168Ah battery bank, we would want to ensure the batteries could be fully recharged in no more than 4 days of full sun, so:
(168Ah/0.80) + (4 Days * 24Ah) / (4 Days * 4 Hours) = 19.125Ah * 48V = 918W
So, for this level of protection you would need a minimum of 1kW of panels and 210Ah of storage. If you plan on using lead acid batteries, you would want to ensure that you never discharged them more than 50%, so you might use two parallel banks of 4 series connected 210Ah batteries (total of 8 batteries). If you plan on using Lithium Ion batteries rated @ 3.6V & 2.5Ah, you would use 84 parallel connected banks of 14 series batteries (1176 batteries).
The cost of the system is extremely sensitive to actual power consumption and the fault tolerance duration.
For Zero Fault Tolerance, 14 * 2.5Ah Lithium Ion Batteries in series charged by a pair of 180W solar panels might be built for less than $2,000. For the 7 day fault tolerance system, the cost would certainly exceed $15,000. Cutting the power consumption by 25% could easily cut the cost of the system by 50%.
Hope that helps,
Fish