If nothing else, cycling teaches you to be resourceful. I've had to scour roadside debris in order to find suitable tire boot material for holes worn thru the tire cord. I've used the electrical tape I use to finish off the handlebar tape to wrap around a punctured inner tube when I've run out of spare tubes (I now carry three spare tubes, two of them with the longer 60mm valve stems). On the fixie, I use a spacer to push out the cog on a threaded hub in order to get a perfect chainline (normally you'd use a threaded adaptor). I've stripped the threads on the hub, so have merely removed the spacer and threaded the cog back onto the remaining threads to get home (with a slightly misaligned chainline).
But I've had to take the bus home for broken chains (when I didn't carry a spare master link) and recently, for exploded rims (braking surface worn down enough that tire pressure blows the rim wall outward). This last one was spectacular - it was on the front wheel and it took the left front brake pad with it! Close to one-fourth of the rim circumference was blown off! But the bike was still (somewhat) rideable. Fortunately, it was on a commute home from work, and all the buses in Greater Vancouver have bike racks mounted in front. And I happened to be very near a bus stop on the line that runs past my townhouse. And I always carry a book of transit tickets in my rucksack just in case. So why was I even riding a suspect rim? Because it was a Deep V with less than 8,000 km on it, and most rims will go at least 10,000 km before you have to start watching the braking surface start to bow outward. (This is if you live in a rainy area like the Pac Northwest; if you live in California, you don't have to worry about stuff like this.)
Oh, and my last fixie, a 1989 Benotto track bike with a retrofitted carbon road fork, decided one day to self-destruct on an after-work training ride. The top head lug broke gong up a short, gentle grade. I could still ride, but very slowly with the bike wobbling. I had to really pull the handlebars into the frame to try to hold the bike together. Finally made it to a bus stop.
Luis