How to pack an arduino project in a box
I've seen Pelican cases being popular for this kind of thing. There are some videos of people using them: http://www.youtube.com/watch?v=9f8nwwCbfRA and http://www.youtube.com/watch?v=3MGRx9KDDg0

Otherwise, to make a sturdy Arduino circuit without a breadboard, I'd suggest using Perfboard to solder the components in. Check out http://www.instructables.com/id/Perfboard-Hackduino-Arduino-compatible-circuit/?ALLSTEPS for a great tutorial on doing this. It is more work than a breadboard, but not too much and worth it for a very solid Arduino device.