It is very difficult (but not impossible) to get work experience at major airports for the simple fact that the relevant security clearences would take longer to process than the duration of the works experience itself, and most temporary passes can be issued for only 5 days maximum, and need a full pass holder to act as an escort all the time, then it is quite simply not worth the time and hassle for the airport authorities, or an airside company to pursue it.
Although flying clubs have been mentioned, and can be great fun to work at, why not think outside the box a little? there are many small companies that carry out contract engineering work for major airlines and aircraft manufacturers. Airline HQs aren't always airside either, and could be a possible avenue.
Good Luck!
