Can an object appropriately isolated from its surroundings become colder than its surroundings? Consider a sealed box, well-insulated on all sides, except for the lid which is transparent to infrared. An object is placed inside the box and the box is evacuated (purpose being to thermally isolate the contents of the box from its surroundings). The box is placed outdoors (in an everyday atmosphere) on a clear night. Let's assume that at the start of this experiment, the box and its contents are in thermal equilibrium with its surroundings. The object inside the box will radiate infrared according to its temperature, which should escape through the lid of the box. With nothing but clear dark sky above, I assume there is nothing to radiate appreciable heat back into the box and maintain the object's temperature.
Question: will the object cool below the ambient temperature outside the box?
 A: Yes, this works.  It's called radiative cooling.  This phenomena has been known for a long time, considering the ancient Egyptians used to make ice this way.
Ideally, something open only to a clear sky would "see" the temperature of space, which is the microwave background radiation.  In practise there is enough stuff in our atmosphere that radiates so that it won't get anywhere near that cold.
Try it some time.  You can do this experiment yourself fairly easily.  Hollow out a bowl-shaped depression in a block of styrofoam, then paint it black.  fill the bowl with water, but leave a little room at top, then cover the top with plastic wrap.  There will still be some conduction thru the air between the bottom of the plastic wrap and the top of the water, but this is still a more effecient setup than the ancient Egyptians had.  It doesn't have to be perfect to clearly show that the method works.  Place the contraption outside on a clear night open to the sky with as few other objects around as possible.  It won't take long for the temperture of the water to go below the air temperature.
A: 
With nothing but clear dark sky above, I assume there is nothing to radiate appreciable heat back into the box and maintain the object's temperature.

In this case there wouldn't really be an "ambient temperature" though.
To elaborate: The inside of the box would cool down until thermal equilibrium is reached between the inside and the outside. At this point both systems will be at the same temperature and until one of the systems change they will remain there. Your assumption that there is no radiation to heat the inside of the box effectively means you assume the exterior to be at absolute 0.
A: There is a well known sinkhole in Utah that displays similar characteristics to what is described in the [nice] accepted answer. There is  relatively little standing between that location and the upper atmosphere Peter Sinks Utah

During calm cloudless nights, this high elevation basin dissipates daytime heat rapidly into the atmosphere. Cool dense air can then slide downwards towards the basin floor in a process known as cold air pooling. Consequently, extremely low temperatures can occur, particularly in the wake of arctic fronts in winter.

The second coldest temperature recorded in the contintental United States happened in that location in February 1985The conditions of a still, clear sky in the desert and the overall extreme cold front led to a reading of -69 fahrenheit.  In this case we are describing the air temperature and not surface temperature: but similar considerations apply.
I remember hearing about that measurement when at Purdue U. in a relatively balmy -20F.
A: On a chilly cold night objects like wood ,straw and grass ( which are poor conductors) don't get appreciable amount of thermal energy  from ground in contact.But these objects keep on radiating energy and are thus NET RADIATORS and hence become colder than the air. Due to this frost is formed on these kind of materials even though the temperature of surroundings air has not gone down to freezing.And of course in such an observation  Frost is colder than the air .
