Takes too much power. Even bicycle gear train powered flight is marginal.
Edit in response to comments: Power here refers to human power. The power (energy/unit time) is a requirement whether it's muscle flapping a wing of feathers and skin and bone; its legs pumping a bicycle frame spinning a prop; a tightly wound rubber band, or a gas engine. The remainder of this argument discusses the power envelope requirements, and ways where a borderline condition can be nudged in the direction of reasonable possibility.
Power requirements: Slower is better. Drag goes up with $V^2$ But at low speed you need large wing areas. Google 'human powered flight'
Things that can help:
Lower gravity. Heinlein, in "The Menace From Earth" has lunar colonists flying in the air storage caverns. 1/6 the gravity. Generating 30 pounds of lift is a far less daunting task. Add to that, you have a lot less wing to drag around, and drag through the air.
Higher air pressure. More air = more lift for the same wing at the same speed. First order approximation is linear. So at 30 psi you need half the wing.
Lighter bodies. The ideal for this would be the physique of a teen age boy with only a few percent body fat.
Fast twitch muscles. This allows brief spurts of very high power. People have a combo of fast twitch and massive recruitment can get all their muscle fibers cooperating. Remember that scrawny kid in high school P.E. who could do 25 pullups?
Drugs. Mechanisms to increase metabolic rate. Brings up another problem -- cooling. Wear a special suit and carry a water supply to wet it to have 'artificial sweat'
More drugs. Topical steroids to increase size of specific muscles.
Surgical alteration. Move muscle attachment points to get better leverage.
More surgical alternation. Exit valves at the bottom of the rib cage so that you exhaled air out the bottom. This should increase lung efficiency by 25%, and reduce the energy needed to breath. Indeed at the right speed it may be sufficient to open your mouth and become a flow through oxygen strainer. Not sure how you would keep your lungs from drying out.
Even more surgery: Replace bones with artificial bones. (Harder than you think. Bones are very strong for their mass)
Glucose IV. Might be able to increase stamina this way. Less demand on digestive tract.
Even with all this, I think the best you can hope for is the equivalent of a human powered sail plane or hang glider.