I am looking for a cheap and easy way of creating a small electronic device that is capable of detecting whether or not it is inside a Farraday Cage.
Hopefully, without an external sensor/emittor.
A simple radar should give you some idea.
If it's inside a small enough cage, possibly a near-field effect-on-a-tuned-circuit type of system would work, instead of needing a pulse-type radar.
This may not be a perfect indicator - it could be hard to tell the difference between being in the midst of a number of conductive planes, and being inside a complete faraday cage. And a cage with a perfectly absorbing inner liner would presumably look like free space.
Assuming you are somewhere in an intact industrial society, detecting existing external sources (broadcast radio, mobile phone networks, etc) is probably more practical. Even absent that, there are cosmic radio sources you should be able to see if something isn't in the way (absurdly ionized atmosphere?)
Bear in mind that what qualifies as a Farday cage is a function of frequency; you would need to use test frequencies comparable to the bands for which you need to know the answer.