Anyone can start their own voice over recording business if they have a computer with an internet connection. Now, notice that I said anyone can START this business. Whether you succeed or not is totally up to you.
I'm not going to promise that you'll be on TV or the radio. But if you have a thick enough skin, metaphorically speaking), and a lot of persistence and patience, you can make some money as a voice-over actor, or "voice talent" as they are frequently called in the "biz."
Again let me state that this is not a get-rich-quick (or even at all!) scheme. Like anything in business, you succeed with good quality and service and hard work. Voice-over work is no different. In fact, it is scary competitive. But since I managed to break into it, pretty much anyone can, as long as they don't have a voice that actively repels people!
Okay, so here goes. First and foremost, you need a product. That product is your voice. Assuming you have one, the next thing you need are demos of how you sound. You do that by recording yourself doing narration of several different kinds. After that you need to market product. At the end of the day, that covers it! Hmmm, sounds like almost any business plan doesn't it? Funny thing, that. But I imagine you want a bit more detail.
How do you record your voice demos? Well, commercial recording studios will do this, if you have the budget and the time for that. There is another way, and you can start today as long as you have a computer with a sound card, and internet connection, and a microphone. Yes, you will need voice recording software, but you can get that for free!
That's right...free voice recording software is available in the program called Audacity. This open source program is amazing. If you don't have a microphone, I recommend picking up a USB microphone JUST to get started. You can get one for about $25. And you can't even rent 1 hour in a commercial recording studio for that!
Once all the above stuff is in place, go directly to Voices.com. There are other such sites, but my success came from there. Sign yourself up, and you'll start getting notified of available jobs every day. Put your demos on your Voices.com site, and start answering auditions. I had to do 100 before I got my first job! Remember what I said about persistence? Voices.com also has tutorials and guidance on how to do everything you need.
If you want to upgrade your equipment, or learn helpful recording and editing techniques for your new voice-over career, you can find hundreds of articles here on Home Brew Audio. Upgrade your home recording studio a little at a time until you find the right level for your needs. Get started with out 5-part series "How to Build A Home Recording Studio."
Now all you need to do is continue learning and continue auditioning. Good luck!