A bicycle definitely can pay for itself, if bike miles replace car miles.
Once started on that path, though, it takes vigilance to avoid accumulating more and more bikes, and upgrading the parts every year, spoiling the economic benefit. ;^)
My father had a "system" to negate this same trend in himself. He decided on an allowable "cost per mile" for his bike habit, which for him was ten cents a mile....and then simply added this amount to a jar in our garage for every mile he rode. Any bike related expense came out of this fund, new tires, new bikes, new batteries for his cyclo computer, etc.
He said this really made him carefully consider the cost/benifit of each purchase and each extra part...and made the eventual upgrade from his "mountain bike from grade school, to my sweet ass speed machine" racing bike, that much more of an accomplishment in baddassity.
On a personal note. One of the greatest moments of my young adult life was when I brought my own bike to dads place for a repair that was beyond my skill level, and he informed me with sadness in his eyes that there was nothing he could do, and that my bike was now not even worth scrapping for parts. Strapped for cash at the time, it was tough to hear that I would now buy a have to buy a new bike....However, my dad simply took his jar off the shelf and without a seconds hesitation, he took me out and bought me a much, much, much nicer bike than he has ever owned himself. He even splurged for an extra water bottle cage and a few fancy water bottles to fill them. I will always remember the look in the cashiers eye as he paid for my shiny new "sweet ass speed machine" with a mountain of change, a bunch of 1$ bills, a handful of 5$ bills, and even a few crisp 10$ bills (earned from the 100 mile plus rides he had taken over the years.) Only he and I knew that each dime on the table was a mile he had spent on the road.
We still ride together regularly, but his jar is filling up a lot more slowly these days. However, by his math, even after accounting for expected replacment parts, maintance costs, ect on his now 25 year old bike....he is only about 3000 road miles away from a "real deluxe tandem set up with a baby trailer and a few helmets" hes been eyeing for me and my family. From his track record, I think the only real question is....why does he think I will be able to get married and start a family before he fills that jar.