How many cubic feet can 1 peltier cool efficiently?
Peltiers are very, very inefficient, compared to a compressor at least. With that volume box, you need to play hard-ball with physics, to get any useful performance at sensible power.

1) Insulate the box properly. Use PU panels (Celotex, Kingspan, that sort of thing), the thickest you can aesthetically use (my 20L coldbox uses PU 70mm thick). Rubber foam sound insulation will not provide anything like that performance.

Once you've figured out the surface area of the box, and the thickness of PU you're using, you can calculate how many watts you'll need to keep it below ambient, and so calculate how many Peltiers you'll need.

Here's an excellent online calculator, which given the power and the delta T you want to achieve, will give you figures for the lowest power, and the smallest (unfortunately not the same) way of achieving that.

2) Reduce the hot side temperature of the Peltiers by using an evaporation cooled water circuit (look up Bong cooler) to waterblocks to reduce the delta T the Peltier has to provide. Depending on the humidity, you will be able to get many degrees below the actual 'dry bulb' air temperature, approaching the 'wet bulb' temperature.

Better still, use a compressor solution, which is head and shoulders above Peltiers for efficiency and capacity.