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.
The best example I can think of is the "Peggy," A Light Emitting Pegboard Display. It is a 25x25 LED matrix display driven by an ATmega168 (which is pin compatible with the ATmega328)
The wiki page has a lot of good information. Including a detailed schematic.
There are few things to notice in their layout.
For one, they use a row common anode setup. That is the current source is on the row, and sink on the column. You have yours in row common cathode. There is nothing inherently right or wrong with either layout. Just something to keep in mind when designing your circuit. If using discrete leds, it just means flipping the led connections. If using a prebuilt LED matrix, it is something important to know. (I'll assume you can easily swap the order to match the peggy schematic. If not, just swap column for row in your head)
They use 74HC154 4-16 decoder/demux chips for row select. Since you only need 10 rows (or cols) you can get away with just one. Of course, there is the issue of current. In your case, at 10 x 30mA = 300mA minimum. To solve that problem they used 2STX2220 PNP transistors which will be able to source up to 1.5A per row. A bit over kill in your case. Since you will just use these as row select switches, just about any other pnp transistor that can source your max current should work just as well. Take a look at Transistor Circuits to figure out what resistor values you'll need for full on/off operations.
On the Peggy board, for the column sink driver they use an STP16DP05. But I have found these difficult to find and expensive. There are many other alternatives like the TLC5916 These use a serial input, and can be easily cascaded. If not, a digikey of mouser search for led sink driver will yield many results.
Alternatively, since you already have ULN2803 arrays, you could use two of these with a single current limit resistor per column. That's a lot of pins, so you'll have to get creative, but it could work for the column sink as well.
Avago published a nice application note titled "Introduction to Driving LED Matrices". It covers this and a few other things.
Best Answer
I would start the design by looking for off-the-shelf LED driver IC (ie. google "led matrix ic").
For example the LE171596A allows you to control 96 LEDs, so you would need ~5 of them. The chip costs around $9 at Mouser.
The driver allows you to turn on and off a single LED (or groups of them), so in order to display anything meaningful you need to update the driver registers all the time (think: just like dynamically driving a 7-segment LED display). To achieve that you need a fast MCU with SPI and DMA. It may be too much for a classic AVR (like ATmega328), but an Xmega could do it. Also pretty much any Cortex-M with DMA is suitable for this task.
Once you figure out what to build (the components) you need to lay them out on a PCB. I guess that 4-layers will be the minimum to get any decent spacing between the LEDs. Fortunately you can order them easily for China.
You've chosen THT LEDs - this brings the problem, that if you pack them tightly there will be no space for other components on the other side of the PCB (capacitors, driver ICs). SMD LEDs are much more suitable for matrix displays.
Finally you also need a low-voltage high-current power supply. These are also available off-the-shelf (eg. MeanWell). If the voltage is too high the drivers will dissipate too much power (and go into self-protection or blow up).