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.
Since the output frequency of a full-wave rectifier will be twice its input frequency, making flicker a non-issue, the easy way is to use a full-wave bridge with no smoothing (since its output frequency will be either 100 or 120 Hz ) and to connect the LEDs in series with an appropriate ballast.
For example, in the schematic, following, 120 volt 60 Hz mains are stepped down to about 12 volts RMS and used to drive a single LED through a 420 ohm resistor.
The transformer puts out about 17 volts, peak, and there are two diode drops across the bridge, so that leaves about 16 volts peak, which is about 11 volts RMS, across D5 and R1.
D5 drops about 2.5 volts RMS with 20 mA through it, which leaves 8.5 volts RMS across the resistor.
So, to get the value of the resistor, we can say:
$$ R = \frac{E}{I} = \frac{8.5V}{0.02A}=425\Omega $$
For more LEDs, you'd change the transformer's secondary voltage and possibly the bridge diodes and adjust the ballast to suck up whatever excess voltage was there that the LEDs didn't need.
Just for grins, Here's the LTspice .asc file you can run to play with the circuit if you want to.
Best Answer
This is pretty close at 58cm wide, although only 10cm tall.
However, assuming board(s) will be mounted on something, why not use two (or more) pieces of veroboard? You could bolt/epoxy/solder some together.
Most PCB manufacturers can make you a board this size if that is an option.
The 60cm sounds about right for 2.54mm pitch, but how do you calculate 360 holes needed for 114 LEDs? (are they 3 leads?)
Things like a rough diagram of your intended setup, type of LEDs used, picture of VeeCAD layout might be helpful to find the best solution.