There's a law of diminishing returns in running; each mile brings less benefit than the one before it. Similarly, running every day is not appreciatively better than running 6 days per week and running twice a day brings almost no added benefit. If running high mileage is a form of insurance, then low mileage is efficient but gambling.
In the 1920's Glenn Cunningham ran a 4:04 mile on 15-20 miles per week. Any high school coach can tell you that it's possible to get good results from runners who do almost nothing (and yes I was one of them). The more talent you have, the less training you need to reach any goal; you might have to train extremely hard to run a 6 minute mile - if you could do one at all - but most men on college track teams could take a year off and run one that fast on almost no training.
The rules for low mileage training are simple: Run hard when you do run (but not too hard), at least twice a week, preferably three or four times per week. Run hills frequently. Run long once per week (but not too long). When you feel good, race frequently; a 5K every weekend works well. When you feel bad, back off and don't run fast or long and don't run hills for a while. Train as specifically as possible.
Beware the testimonials
It's not hard to find people who claim to have run good marathons and ulramarathons on low mileage, but you have to look closely at their training to see how they do it. One man claims to run 100 milers on 15 miles per week, but if you think about it, if you ran nothing for 7 weeks and then ran 100 miles you'd be averaging 15 miles per week, so he's not counting his races and he actually runs 45-50 per week. Others do a lot of other aerobic (and anaerobic) exercise besides running and they're not mentioning the cross-training. A third common claim is "I used to run 70-100 miles per week and now I'm running even faster on only 20 miles per week;" the endurance they gained from the higher mileage takes a while to be lost and now they're rested - check to see how they do in 2-3 years.
Jack was the first of the new breed of run-less runners. He was a world-class cyclist who retired at age 33 and took up running; he ran a 2:11 marathon at age 41 and 2:20 at age 50. His training is usually described as "three days running per week with some biking on the weekend." His typical week actually was 20 miles in two hours one day, 15 miles in 90 minutes on two other days, one of which included 9-12 times 1000m at race pace with 600m recoveries (3 minutes on, 3 off; still averaging 6 minute miles) and his biking was a century done in 3-4 hours. The 2:11 was done on 70-80 miles per week, before he switched to the other schedule.
Foster's cycling leads to cross-training. Many runners find that running every day leads to injury or burnout, but that they can do a roughly equivalent workout using some other aerobic activity. Yiannis Kouros is secretive about his training, but reportedly does a lot of rowing. Geoff Roes, who has been nearly unbeatable in ultras recently, does a lot of biking and hiking in addition to his running. The idea is to strain the circulatory system, but use different muscles or use them differently, decreasing the repetitive stress.
The Tabata Fad
The current obsession with low-mileage comes from an experiment that showed increases in "anaerobic capacity" (at best a nebulous idea) doing 8 repetitions of 20 seconds of all-out effort followed by 10 seconds of rest were better than increases done by steady exercise. There are a great many flaws to the study and people are stretching the results to match whatever they're promoting; people will buy into anything that says they can do less and get more.
[I fully expect to get a bunch of angry responses to that. So, in advance: screw you, too.]
Why 20 seconds? Because one uses a creatine phosphate/ATP shuttle for energy for a few seconds, but only world-class sprinters can use that for more than 15 seconds, so there's a small oxygen-deficit measurable in ADP concentrations. Why 10 seconds? because it's an artifact of earlier studies of oxygen uptake that used a 2:1 work:rest ratio, which makes sense energetically when dealing with minutes and not seconds. Why 8 repetitions? because 8x20 or 160 seconds is about as long as most people can stand "lactic acid build-up" - another misnomer.
The study's actual finding was: training anaerobically helps one train better anaerobically.
Ultras on low mileage
It's possible, especially if you're already in good shape. The current Crossfit craze has one thing going for it: for experienced ultrarunners, the limiting factor tends to be muscular fatigue. Crossfit does a good job of stressing the smaller muscles usually used for balance and posture (as do yoga and pilates, though in a completely different way) and these are the ones that tend to fail. Standard running training don't stress these muscles much, but hills and fast trail runs do. I prefer to run; it's more specific to running.
Next up: peaking strategies