A resounding, "No" on all levels, unfortunately.
Firstly, fiber strong enough does exist, but its considerably larger than what is depicted in the movie, and also massively expensive. Secondly, materiel strong enough to make the actual slingers that small and strong simply does not exist. Nor does technology to effectively throw fiber like that, let alone something as sticky as depicted in the movie. (Not to mention something that sticky would be ungodly to coil)
But, even if through several technological miracles, you managed to actually create effective slingers, you still couldn't use them, since, in the best case scenario of a full tension, gentle swing, it would chafe your wrists something awful. If you jumped off a building and used it to catch yourself, best case scenario, the straps cut right through your wrists and you live till you hit the ground. But depending on angles, it could rip off your arms at the shoulder, which would probably lead to bleeding out before even hitting the ground.
Why did you have to make sense of the mechanics of Spider-Man? It was a guy who got super-strength from being bit by a spider. By the way, he used wires with liquid cement on the end in the first comic, he also had web wings. SENSE WAS NEVER THERE.