Step 1: Think first
I personally needed it to fit behind the passenger seat in my car. This limited the dimensions, so I had to find a size that would work with different arrangements of beverage containers and ice containers. I chose an internal box size of 7W x 10.6L x 10H (inches). Different applications would require similar planning.
I considered a few materials for the insulation: FIP polyurethane foam inside a cardboard box, EPS (beaded polystyrene foam), and extruded polystyrene foam. Since the first two ideas suck and I already had a sheet of 1.5" pinkboard, I decided to use that, thus giving the box its absurd 3" thick walls.
Step 2: Cutting things, building things
All the panel sizes were cut on a radial arm saw, though one could easily do this with a circular saw or a hot-wire cutter. I could've done a bit better on a few of the pieces, but the biggest problem wasn't my measurements. It's that the sheet was bowed and all the edges were not perpendiculars, but radials. Make sure to get a flat sheet, especially if your foamboard is thicker.
The box is assembled using a polyurethane glue such as "Gorilla Glue" or any of the similar glues on the market. The advantage here is that they have a good bond strength and they expand to fill any gaps between the sheet edges. The assembly of my box was made difficult by the bowed panels as they would constantly leave gaps at their edges. That's what clamps are for though. The clamps are pretty necessary anyway, as the polyurethane glue doesn't have a very high initial tack. If your panels are flat, you could probably get by with using tape to hold it tight.
Step 3: Foil facing
For the sake of durability, the aluminized face should be positioned toward the inside. You can identify the aluminum face of the film by rubbing on it with your finger (the aluminum comes off) or using an ohmmeter.
The mylar is attached to the box using a spray adhesive. I used green label 3M Spra-Ment, although there are several compatible adhesives available. CHECK THE LABEL. Not all spray adhesives are safe on foamboard. Cover the box by rolling the sheet around the sides, leaving the excess to form flaps to cover the top and bottom. If you can wrap a present, you can do this. It's best though, to spray the box (not the film) one side at a time. The glue tackifies quickly, and you can't reposition the film on tackified glue. Once the film is in place, It might be a good idea to use clear packing tape to hold down all the edges and reinforce the corners.
Step 4: RESULTS
To base my expectations, I did a rough calculation involving the wall thickness, radiant area, thermal conductivity of the foam and the fusion enthalpy of water. Given that the temperature in the vehicle swings from 65dF during the night to as high as >140dF during the day, I found an average of 100-105dF to be reasonable for the calculations. The results indicated that the box should be able to keep 2kg of ice for around 27-30+ hours. Increasing the box dimensions independent of wall thickness will reduce this time. Increasing wall thickness and the mass of the ice charge will increase this time.
After running the box in the car for a few weeks, I can say the calculations are pretty accurate. The ice blocks come out of the fridge at 14dF. With 4kg of ice, it can easily maintain acceptable temperatures for four days even when it's opened four-six times per day. I just rotate different sized ice blocks (frozen beverage bottles and 1/2gal milk jugs) to accommodate different container arrangements and expected durations. It might be a good idea to make your box fit available rectangular ice bottles. Round ice blocks waste a lot of room.
The coldest I've ever had the bottom of the box (coldest part) was about 19dF. The warmest I've ever had the top of the box (warmest part) was about 42dF. Such high temperatures are only reached when the box is in the sun, the ice charge is growing small, or the lid has been recently opened. It averages between 34-38dF most of the days.