There is literally no other country in the world that could have this mess take place. Like, anywhere else they'd just have another election to resolve the crisis, or respect the result of the last one.
So admittedly Obamacare is on average more unpopular than popular at the moment. However, negative opinions of it are not actually topping 50%, so it's not even like a majority of people are that bothered about it, let alone the "nobody wants it" rhetoric of the GOP's right wing. But in a sense that's not the issue; the issue is that if you don't like the decisions of lawmakers and you don't like their laws, you take that argument to the country and win the debate. That's how representative democracy works. The Republicans in 2012 categorically failed to do that. If Obamacare is the most evil thing on the planet ever, the GOP has to admit that it failed to make that argument when it mattered.
The solution to the problem is categorically not to just stop paying thousands of civilian defence staff, passport office staff, museum staff, national parks staff, and other people who are keeping the US safe (the former two cases) or contributing to its economy which isn't in great shape to start with (in the latter two cases). Silly people.