Odds of Getting In
While advanced technology and the ability to scan data electronically has certainly cut down on the need for bike messengers since their heyday in the '70s and '80s, certain legal documents need to be hand delivered and have their originals signed. While UPS is fast, car couriers can't compete with NYC rush-hour traffic like a cyclist can (source).
Also, larger items like wardrobe deliveries or packages that must get across town ASAP need bike couriers. There's not oodles of people who are willing to put their lives on the line for minimum wage—and who actually have the stamina for the job. So yes, there's work out there if you want it.