Winning GC without ever winning a stage?

It's all about cumulative time. Consistently finishing well gets you that. You can win several stages and still be far back.

Mark Cavendish won 5 stages in the Tour de France this year (the most of anyone) and still finished 130th, 3 hours and 15 minutes back on the general classification. That's because he lost a lot of time on the mountain stages. He's a sprinter and wins flat stages. The big time losses he had in the mountains added up to a lot. On his stage wins, the peleton behind him was usually close enough that they got the same time, so while he won the stage, he didn't gain any time on the overall. On the mountain stages, he would often lose the stage by more than 10 minutes; sometimes a lot more.

Cadel Evans won only one stage in the Tour de France this year, stage 4. He didn't lead the tour until stage 20, the next to last stage. He came in second on that stage but took the overall lead because he was much further up on overall time than Tony Martin who won that stage and who had had some very bad days in the mountains. In fact, 43 riders who were slower than Martin on that stage were still ahead of him on overall time. Evans finished high up in time standings every single day among the riders who had a chance of winning the overall. There were some days where breakaways took some time out of him but they gave it all back eventually. They gave it back in the mountains and/or the time trial. Consistency wins. It wasn't the flashiest way to win the Tour de France but it was tactically excellent.

Last year (2010), Alberto Contador did win the Tour de France without winning a single stage. Again, he finished well every day.

Yes it is possible. If there is a group of riders that cross the finish line they all get the same time. You don't have to be the first across the line. On the following stages you can be in a different group of riders then the ones that you finished with the day before. There are riders that do very good on the mountain stages and others that are sprinters. The GC riders are pretty good all around riders. Some of the top sprinters can lose several minutes a mountain stage.

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After three weeks, a race leader may only have a few seconds lead over his rivals. This few seconds may be made up in just one stage, or lost in one stage.... propelling a second stage winner onto the podium as GC champion.

It is like the decathlon in track and field. You may become 'the greatest athlete in the world', not by winning the events, but by being the best overall in all ten events.

Soccerrref