The thing you have to understand about contracting (and really any paid work, I've noticed the same effect as a big-company employee) is that people will pay a large premium for a track record of delivering working solutions. When you first start out, your asset is "I know this programming language", your clients just want someone who knows that programming language, and you're competing with all of these people on oDesk and eLance who also know that programming language. After you've delivered successfully on your first couple projects, your asset is "I delivered on these projects and it made my clients $X", your clients want someone with a track record of delivering on projects with minimal handholding, and you're competing only with other people who have had wins of similar magnitude. The supply of these is much lower and the demand is much higher, and so you can charge much higher rates.
I would avoid oDesk type rates at all costs. You're better off going to small business networking groups and making simple brochure sites and wordpress installs for $1000-$2000 a pop.
This is just my opinion and I can only speak from a US perspective, but yes and no. The ones who are able to do this have already gotten a 10+ year head start and have built up a deep network from which to source work from. The ones who are starting out today are more than likely having a very tough go at it. Competition for contracts is much more intense compared to 10, 15 years ago and you really need to bring more to the table than just "javascript developer" or "wordpress consultant".
I know there exists this love-affair on HN with running off to become independent contractor, live on a beach writing code and making your own hours. I'm not saying it's impossible, but the reality though is that in the end you are building a business (even if it's just you). All the energy, time, risk and money that it takes to run a regular businesses will be just the same with running your own consultancy. The downtime you think you will have to yourself in-between contracts will be eaten up by having to find the next contract. If you happen to unfortunately live in or near a city with a high cost of living, the pressure will be even more intense.