There are mustachian ways to fly. I disagree with the argument that pleasure flying (or even utility flying - such as, to do a weekend commute) is automatically facepunch worthy; although not cheap, there are ways to do it at reasonable expense.... Now, the path is not extremely straightforward, but if you "hack the problem" intelligently enough you can pull it off.
There are two main barriers to entry.
1. Quals (getting certified as a pilot)
2. Iron (finding and paying for a plane to fly)... impacts #1, because usually you have to rent a plane hourly to get training
#2 has a lot of hidden costs: hangaring, insurance, maintenance for example.
Getting someone (like the USAF) to pay for your training and lots of flying is a great way to do it; but not open to everyone.
The other best way to do it is - don't go it alone. Find a club or a community. Hang out at the local airport, be an airport bum.... get to know all the old guys who work on and fly their planes on the weekends (hint - many of them are or were once instructors, and many are either licensed mechanics or built their own planes). Help them turn wrenches, wash airplanes (there is a reason they are called bugsmashers!), be a spare hand helping someone rebuild engines or install avionics, haul stuff around the airfield, do odd jobs. You will learn lots along the way, and in turn may get some flying time. It's a long game - it will take you a while; but in addition to getting a little flying you will build relationships and friendships, and learn a lot from the experienced old heads.
Learn about experimental aviation. Don't get thrown off by the word "experimental"... yes, there are idiots who build and fly deathtraps, but there is also a HUGE community of experienced and very skilled airplane builders and pilots in the US, and many well proven designs - some of which are more efficient, better performing, and *BY FAR* less expensive to operate and maintain than certificated types. Stroll down the shade hangar at the local airport: most of the old, beat up, rusted airplanes with flat tires and leaking oil fading away in the sun are the certificated types. The experimental ones are typically much better maintained (and more often flown!)
Glider/soaring clubs are probably really the only true mustachian flying answer. In addition to all the club/community benefits I mentioned above, you get to fly super cheap if you find a good club. As always, getting instruction will cost you some, but pound for pound and hour for hour gliders are cheaper than any other planes. You still learn airmanship skills that translate directly to powered flying should you take that step. For example, there is a club near me that requires $65 dues and 4 hours of duty, per month. For this you get free access to all the gliders, and only pay $1/100 feet for towing. On a good summer afternoon you can fly 4-5 hours for 20 bucks. By comparison, most powered light aircraft rent for well over $100 per hour, counted any time the engine is running (even during taxi), and that doesn't always include fuel.
Beware of the pitfalls of the money furnace: hangaring and insurance are big ones; maintenance (most especially for certificated types) is another. Fuel isn't cheap - if you can find an airplane that can run on car gas (learn about whether/how ethanol impacts your engine) you can cut fuel costs almost in half. By joining a club, you eliminate (or at worst, share) many of these costs.
Long story short, it is possible, but there are tons of things to learn and it's a very gnarly problem to figure out. It is an entirely different problem for every person and every location. But figure it out in a way that gets you into friendships/community, and you'll have a lot more fun while not spending as much as you would by just diving in alone.
