Learning By Doing Theory

There's little demand for inexperienced workers. Almost all companies want to hire workers with prior experience in their field, to minimize training expenses, and to start productive working immediately. This leads to the extreme environment I have been seeing for the past year where many companies won't even bother with a person who isn't already working in their field, and all companies demand prior experience. Those with little to none, like myself, where left with a catch-22 of requiring the experience to even get the experience required. This had to be frustrating for the companies too, who keep wondering why the figurative watering hole has been permanently dried up. (Like most professions, IT workers like me do not spring fully formed from Zeus's forehead!)
I think the entire system would benefit from workfare. People would be paid to do things, thereby getting the money to raise them out of poverty, but more importantly, they would have experience that would make them more attractive to hire in the first place. Companies would gleefully hire up the workforce bunch, who have prior experiences doing what they want and can be laboring productively starting immediately, and the training expenses have already been handled. The paying government would also benefit of progressively shrinking welfare rolls as the petitioners get snapped up into the workforce and the economy recovers.
However, for this to work, workfare would have to revolve around all sorts of industry's work, without stepping on that industry's toes by competing. We do need shovelers and road crews to fix our roads, but having done that doesn't benefit any industry. If the government did manufacturing, engineering projects, and IT, three industries with large demand for the foreseeable future, wouldn't companies that do manufacturing, engineering, and IT complain about the competition? (Yes, undoubtedly.) So the government would have people manufacture build and program systems....that would be thrown in a hole and forgotten. Not good for the morale of the workers, nor would the government really benefit from the results of that.
A better idea are institutions like NASA. NASA has a massive crew of engineers with numerous skills that almost any company would want to hire, but it focuses on something that companies can't yet do profitably for themselves -- space travel. (Space engineering is extremely expensive, and won't be profitable until we figure out some way to keep the expenses down.) So a workfare engineering facility would have to manufacture things that are useful to the government, but not profitable for private companies to make. Specialized tools for the CIA, perhaps. And government IT would have to focus on programming and maintaining computer systems that are somehow unprofitable for private contractor work. I can't think of any of these offhand, but I'm sure they exist and the NSA readily knows what these sorts of things would be.