i agree with what LM said. when i haven't been lifting and feel weak it does suck. but when i haven't been running and get winded easily just doing simple things that require endurance that's when i really feel out of shape.
someone quoted an article saying the 4 components to a healthy body are strength, power, muscle mass, and endurance. and claimed that running only helps endurance and actually hurts the other 3. not true at all!
obviously running helps endurance much more than lifting does. but tempo runs, hill workouts, interval training, speed workouts are all also parts of running. if you are a distance runner you should be doing all those as well. and those cover the strength and power aspects.
and as far as muscle mass, well that has nothing whatsoever to do with fitness. that is simply the aesthetic outcome of lifting a lot, it's the byproduct of eating tons of calories and doing lots of strength work (and as someone earlier mentioned the medical world has begun discovering that simply eating a lot of calories, no matter if you burn it off, is detrimental to the health of the body. and yeah you need to eat a lot to maintain a lot of running, but you need to eat much more if you are doing lots of lifting and trying to put on lots of muscle mass). muscle mass may be a fitness goal because people wanna look strong but it has nothing to do with the actual health of a body. i mean consider women instead of just men. women don't want to build up lots of muscle mass like men do, because they want to be thin, they can still build up strength and power and endurance and be thin and have no need to build up big muscles. muscle mass is purely aesthetic.
now granted running works the endurance, strength, and power of only the lower body (as well as working the heart). so lifting for the upper body is important for overall health and i think most runners would agree that adding some lower body lifting on top of their running increases the benefits to their lower body.
the fact of the matter is that both strength training and endurance training (the best and most efficient kind is, of course, running) are crucial to overall body health. runners don't say that strength training is bad and ruins fitness, or if they do they are wrong. but for someone reason these crossfit types pretend that running is bad for fitness, which is at least as wrong as saying being strong is bad for fitness, and probably even more so because most physical activity humans do is with our legs and is endurance-related (i.e. walking, running). just look at how we compare to other animals of similar size in terms of strength and how we compare in terms of endurance. our endurance is very high in comparison while our strength is very low.
i consider the pillars of physical fitness to be endurance, power, flexibility, and agility. Endurance is obvious as we are engineered as endurance creatures, plus its very important for heart health. Power requires strength plus speed. Flexibility is important for the health of our muscles and contributes to agility. And agility requires flexibility, power, and balance. if you are looking for only one type of workout to do for overall fitness i think its gotta be martial arts. while apparently here in the West we are still arguing over strength and endurance, the East had fitness figured out eons ago. Martial arts requires superb strength and speed to create power (without having bulky muscles that impede movement), flexibility, agility, balance, and even endurance to keep it up for extended periods of time, plus on top of all that it requires mental focus!