Computers have about killed off the bike messenger business, even in big cities. There's just no demand for somebody to schlep a document across town when you can email it in five seconds and get a response when the messenger is still running down the stairs to the parking lot. Packages are a possibility, but the market's pretty small.
If you're determined, a high school kid here made a few bucks last summer doing grocery shopping, pharmacy pickups etc for people who didn't want to do it themselves. He'd either pick up an order they'd phoned in or choose the items himself if they gave him a list. Worked well enough in downtown Reno, where you can go to the store in 10 minutes. Not so well as our suburbs expand, though.