Everyone looking to immediately blame this on government should think about what's involved and what probably happened:
1. The contract went to the lowest bidder and/or the firm that could do the most backroom political deals to win. This is not necessarily the team you want doing the work, nor are they necessarily the most capable.
2. It's a huge, monster systems integration challenge. There are probably thousands of XML data brokers, enterprise service buses, web services libraries, and wrappers of wrappers of wrappers of abstraction layers to get the exchange, the insurance companies, the tax records systems used for eligibility verification, the authentication, etc etc etc talking to each other. This is one of the things I do for work on various big systems projects, and it's hard when you have a competent team. When you're dealing with the "offshore delivery centers" of the firm in Point #1 above, it's an absolute nightmare.
3. Every outsourcing contract, public or private sector suffers from the same problem -- it's always more expensive, and the people involved don't have any incentive beyond a paycheck to see it work. I've seen that happen all the time as an FTE in companies overrun by consultants. The consultants don't care what happens as long as they're billing time. If they deliver garbage, so be it, as long as it can be shown that it does what the contract says it does.
4. Continuing with the "don't care" theme, there's also no incentive for the contractor to get it right the first time. Even contracts with penalties for failure or missed dates aren't a big deal because they can bill way more cleaning up the mess they made.
5. I'm sure the "outsourcing partners" weren't forthcoming when the RFP was put out and they saw red flags. Some outsourcers like to trap the customer and have them think everything's sorted, when there's really a huge problem with design/specs/whatever that will mean a very expensive rewrite later on.
6. Any project with a huge red target date on the calendar that is not flexible is doomed to failure. Problems like this lead to stupid things that PMs do like stuff more people onto a late piece of the project where it clearly doesn't help, and it leads to people taking shortcuts to rush it out the door.
7. There was probably immense cost pressure, not from the gov't itself, but from the outsourcer trying to squeeze every nickel out of the deal, and so it probably runs on half the hardware it needs, and has no DR facilities.
8. It was probably slapped together by hundreds of 24 year old new graduate business analysts, hundreds of 30 year old PMs, and thousands of offshore resources of dubious quality. Look at pretty much any bespoke line of business web application you have to use for your job. Chances are you hate it and it has maddening bugs that make it hard to live with. Now take that same code quality and put it in front of Joe Average, and I'm not surprised people are complaining.
I honestly think they should have done this in-house with supplemental hired gun contractors for the areas they needed it in. Despite the stories, I'm sure working for a government agency has its advantages. I would think that people (myself included) would welcome a more stable employment environment (at the expense of salary,) a stable retirement system, and the ability to work on a critical system that affects people's daily lives. The problem is that people see IT people getting rich at Google/Facebook/Latest Social Media Startup and think that they're going to be the next one to make the big time. Reality is that most people are mediocre coders/IT people and they're never going to get a big payday supporting the current IT employment model we have.
Also, this entire mess would have been avoided by extending Medicare benefits to everyone. Doctors would be happy because they would get paid without questions from insurers, patients would be happy because they wouldn't have to deal with insurance companies -- the only people who wouldn't be happy are insurance companies, which is why we have the system we have now. Seriously, the Medicare system processes payments for doctors with very little difficulty -- because we have the insurance companies involved, we had to build a completely new system.