Of course GovNet is possible. The question is whether it's feasible. It would not be difficult to create another internetwork that has no connections to the current internet, and I'm pretty sure some government agencies, like the CIA, already have such things. The GovNet they are pondering now would obviously be much more extensive, but all it would take to implement is people and money.
Would the added security outweigh the disadvantages, though? There are two possible ways to implement this, I suppose. One is having an actual seperate government network with no physical connections to the internet. The other has already been mentioned, something along the lines of VPNs. The problem with VPNs is the same as the current problem. Everything has to be set up correctly to be secure, and it is still not 100% secure.
Would a seperate physical network solve the problems of security? Not in my opinion. Considering the number of people who work for the U.S. government, the number of contractors, and the fact that the vast majority of networks are comprimised from the inside, it seems like the better plan would be to get serious about security with the current situation.
If there is a seperate GovNet, wouldn't it sort of defeat the purpose of the internet? How do you download software, drivers, etc, get online support, and whatever else you need to do? Running two parallel networks, a public one and a private one, with no physical connection seems like more trouble than it's worth for most government agencies. I can see where it may make sense for some, but it seems like overkill to me. Also, if security currently is bad, my Spidey sense tells me it would be more sensible to fix the current situation rather than starting over from scratch and creating a whole new set of problems.
All that said, I'm of course assuming that whatever they come up with will have little base in reality. If they come up with something that seems sensible, then good for them. It is possible but seems unlikely considering some of the government's , and specifically Congress', views on technology. I assume Congress will have to approve funding for something like this.