Converting a simple bed into something that can retain cold air for a long period of time should not be extremely difficult. Actually it should be quite easy to conceive the simple idea in the mind.

What you can do is to cover bed with some sort of a cover. A plastic type tent or a cover should suffice for this reason. The main requirement should be to make the whole thing in such a way that it makes an air tight room for a person to sleep in. It should be made air tight to an extent that cold air should not slip out of it.

One should be careful in making sure that user (the person who would sleep in such a bed) should have ample supply of fresh air in it so that he (or she) can safely sleep in such a bed. This would necessitate us to make sure that proper ventilation is taking place inside the box.

I wonder how this can be done efficiently. But it should also not be difficult to accomplish. Either provision of fresh air can be made from the duct directly. Specially if the air conditioner to which it is connected at the far end is taking in fresh air. In this case you would require to have another duct that takes used air out of the bed. If the air conditioner is designed to recycle the air, then there should be inlet for fresh air in the bed too. This is to ensure that the person sleeping in the bed does not run out of ample supply of oxygen to breath.

Ensuring that the user can breath properly and safely while sleeping is quite important and cannot be over emphasized.

In order to make the whole system efficient it is quite important to use proper insulation inside the bed. Insulation is important so as to conserve energy in some sense. In simple terms, insulation is important so that the ambient heat outside the bed does not enter inside it. Conversely, the cooling achieved inside the bed is not released out of the bed as a result of bad design.

This is where the design comes in. A good design of the bed would not only make the whole system efficient, it would also make it more aesthetically appealing and also comfortable. To this end, I presume many designs can be conceived which have all the necessary elements of a good air conditioning system.