How do I start my own answering service business?

Contact your local government and get a business license. It's 20 bucks here in Washington state, but might be more other places.  Then, put an ad in the paper, in the phone book, etc... Try to advertise where you can.  I personally don't know of any companies that are hiring for this. You might just have to find your own customers.

I have a friend that owns a answering service. There is a substantial investment required for a building (leased) with telecommunications services available. Very expensive computer/network and telecommunications equipment is needed, and high bandwidth communications services will have to leased from the area phone company (multiple T1's or a T3) that cost $1000's per month+ the leased equipment to interface them, like a PBX system, and a very high level of technology expertise will be required to operate/troubleshoot problems. You might want to investigate an answering service business for sale.