Is it possible to shield a camera so as to record from the inside of a running microwave oven? Would it be possible to create shielding for a camera, allowing it record food being cooked from the vantage point of the inside of a consumer microwave oven without the camera being damaged?
Wikipedia says a consumer oven usually uses 2.45 gigahertz (GHz)—a wavelength of 12.2 centimetres. So surely you could create some kind of cage around it which allows light to pass through to the lens but not allow microwaves?
 A: You correctly surmise that you could make a cage with small holes, so that the visible light could pass through but the microwaves would be blocked.
However, you could have some major problems if you decide to put a metal object inside a microwave oven - see the same wikipedia page you got your frequency information from. You would have to construct your "cage" in such a way that it has no points or crinkles to prevent high charge concentrations and arcing. The cage may also get quite hot.
Maybe you could surround your camera with a smooth layer of kitchen foil (perhaps stuck to some nice 3-d former) leaving a small hole for light to come through. You should perhaps also investigate making sure there is no potential difference between the oven casing and the cage (see this answer).
What properties make a good barrier for microwave (oven) radiation?
Anyway - be careful!
A: You could put your camera in an invisibility cloak for microwaves. I don't know how that would distort the images, but it has probably the advantage of not disturbing anything else inside the microwave. If you use a metal cage, there might be interference effects, and the existence of the camera would influence the thing you want to examine.
A: Unless you are using an oversized microwave (like industrial or scientific size) and a large protective shield for the camera, I doubt it would be practical to film the inside of a microwave oven while it is running, without destroying said camera.
If you are only looking to simulate the effect, you may disconnect the magnetron circuitry from the microwave oven. First off, discharge the high voltage capacitor. In most ovens, you will find a circuit board with a relay, with two wires leading to the primary winding of a heavy transformer; the secondary winding leads to the magnetron. You should be able to simply disconnect the primary side of the transformer, from the circuit board (preferably) or from the transformer itself. Avoid touching the secondary side as it is connected to a high voltage capacitor even if you have already discharged it (you might still get a very bad shock from a partial discharge).
Depending on the complexity of your oven it might or might not work (I suppose some microwave ovens would detect this as a fault). If you want to also record the classic magnetron hum sound, do it in a separate track, from outside, before disconnecting the magnetron.
Serious warning / disclaimer:
Messing with microwave ovens (and any other electrical appliance) can be dangerous! Attempt to do this only if you know what you are doing. In any doubt, ask an experienced electrician or electronics professional friend for help.
A: The easiest solution is to use a fiber-optic camera, i.e. one with a fiber-optic connection between the front lens and the actual camera electronics. You can easily bend the fiber (but not too much!). You can now make a small hole in the microwave (smaller than the wavelength, insert the fiber. Bend the fiber and wrap it in aluminium foil.  There will be little leakage through the hole, and the microwaves that do leak will be absorbed by the foil. 
