Create large Arduino powered LED sign
Use Monochrome LEDs, 60mm x 60mm 8x8 RED LED matrix common cathode are low-cost

So each one is about 2.3inches square, with 64 LEDs.
4 high x 16 long would be about 10"x3'

LED drivers are quite complex to program. However, there is an Arduino library for MAX7219.
MAX7219 is cheap, easy to get, IC to drive a monochrome common cathode 8x8 LED matrix.

One MAX7219 / 8x8 LED. They are 'daisy-chained' end to end with a few signals, so it won't be too hard to wire up a large display.

To get up and running you could get a ready made module with 8x8 LED matrix using the MAX7219. These are easy to get from web sellers. Unfortunately, the PCBs I have seen don't 'tile' fully, though you could stack two to get 16 LEDs high.

Kits with a PCB, LED matrix, MAX7219, sockets and components are available to make the same thing. You could practise making them, to see if you feel confident to make a large display.

Buy a ready made module, and a couple of kits (just in case you break the first one), and do some experiments to get comfortable using them.

Then to make a large panel you'll need PCBs. Make them in smaller pieces, maybe one to four LEDs/PCB.

If you design individual PCBs, one/LED, you could use Eagle to design the PCB.
However, 'free' Eagle has a size limit which might be awkward to overcome.

RS provide a free DesignSpark PCB package which has fewer restrictions than Eagle.

There are a bunch of low-cost PCB manufacturers on the web, that you can find with a search, who could make the PCBs for you.

If you have a friend who could make reliably make single-sided PCBs, that might be sufficient for the job (I've done a single sided PCB for something similar). Put the 8x8 LED matrix in a socket, and the MAX7219, in case things go wrong, and you want to use a commercial PCB.