"It's also a significant number of Watts."
Not necessarily - if they are arranged in combinations that would allow many to be in strings running from 1 voltage source, say 100V DC, then one could have 50 LEDs in each string (say to form a single segment of 7-segment display for example), then there would be 46 strings at 20mA each (with typical LEDs and not some high wattage LEDs for example), so 9-10 amps of current.
Times 100V - I guess even like that would be 900-1000W.
Still, no worse than a few light bulbs.

4' x 4" = 48" x 4" = 192 square inches. Looks a unit check is in order to start with. 4 feet x 4 inches, or 4 feet x 4 feet?

http://www.mpja.com/prodinfo.asp?number=18460+OP
4" high x 3" wide, 16 of these would fill a 48" board, 560 LEDs.
String of shift register like 74AC299 down the length of the board. P-channel MOSFET to source current for the rows.
This particular part is common anode vs common cathode, could be driven like this:

Find a P-channel logic level MOSFET that can source 500mA, if find a higher drive part than have it drive more columns.