I had my freewheel fall apart halfway home in a 35F rainshower on a day when I forgot my cellphone. I walked the bike to a freeway overpass to get out of the rain. I managed to rig it up as a fixed gear by using a multi-tool as a lever to pry parts around. (Off topic: IRD freewheels are MUCH nicer than 25 year old NOS originals).

I only had one flat on the road in 2008 (knock on wood). Usually when I get a flat, it's a slow leak and I don't notice it until the next day when the tire is completely flat. Even in this case, it was a slow leak come to think of it. I was 30 miles from home and chose to just stop and put air in the tire every 10 miles or so rather than bother to fix it on the road. So I guess I never really fixed a flat tire away from home this year.

I generally carry a spare tube, multitool, spoke wrench, patch kit, pump, spare batteries, spare light bulbs, and raingear when commuting. On long rides, I usually carry a spare folding tire and food as well.
I only had to call for a sag once in 2008 (again, knock on wood). That was when my key broke off in my bike lock late at night. I had to get a ride home from my wife to pick up a spare key then drive back to retrieve my bike.