Yes
Yes, it is possible.
I've done this (to an extent), and it is fairly simple with the right equipment. To begin with, in most cases you may want to avoid store front businesses, unless wishing to pay a premium. They tend to charge excessive prices for such adapters.
If making a connection from mini dvi (which your MacBook Air has) to RCA jacks, something like this should suffice:
http://www.amazon.com/DVI-I-Compone...7?s=electronics&ie=UTF8&qid=1292622220&sr=1-7
One might cobble a variety of cables together to achieve the desired result, with doing so most elegantly something of an art, with research involved. In my case, I was making a connection between mini dvi to HDMI, since it was a newer HDTV. If you have the option, that should be preferable. Look on Amazon or elsewhere, and you'll find the appropriate cables.
What I never figured out, in only playing with it briefly, was how to achieve any more than video. Video itself was a simple matter, with the only real trick in selecting the proper input through the TV, then also aspect ratio. TVs will vary in this, each different, so you will need to learn what to adjust to entirely fill the screen properly. Once learned, it is a simple matter. The problem I had with audio, or lack thereof, probably relayed back to this older MacBook, whose mini dvi does out output audio. Thus I had to use a separate cable from the audio mini jack (also used for headphones). The problem was I could not figure out how to select both audio and video inputs on the HDTV, as it only wanted one at a time. Probably my error somewhere, and I turned my attention elsewhere. If I had connected the mini audio jack to external speakers, then I would have had sound as well; that would have been one solution.
However your MacBook Air should have the newer version of mini dvi which outputs both video and audio. That will solve a lot of problems. If to HDMI then you would probably experience no problem at all. If to RCA, then probably not, but I'm not sure.
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This said, you may wish something more elegant than having your MacBook Air wired to your TV. Although I have not used it, if wishing to no more than watch iTunes content then the new Apple TV would probably be your best bet, both in simplicity and cost. There would, however, be at least two problems. One is that it offers only an HMDI output designed for newer sets. So you would have to cobble something together if using RCA. If the option, HDMI will provide you the best signal. The more serious problem in this instance is it seems you want to watch something online, not iTunes.
Therefore, Plan B. Something like the Boxee Box might be the best solution. It is designed to access videos, music, etc. from a computer, apparently wirelessly. The caveat that you should research this to determine if it will meet your specific need. Roku will provide access to such things as NetFlix, but it may not stream your own content.
The bottom line is that there are a number of options out there which may work for you. It will just take some time to determine how you will best be satisfied. A good place to research this from the ATV forum on this site:
https://forums.macrumors.com/forums/100/