If Hedge Fund doesn't have the money and goes bust, "what happens"?
Who loses, a brokerage? Individual traders? Government bail out?
Their broker is liable for the shortfall. 'You' don't ever buy and sell shares. Your broker does the buying and selling with other brokers, on your behalf. And they have to guarantee every single trade that they do.
Brokers, like banks, are required to hold a certain amount of capital to make sure they don't go under. If a Broker loses more money than they have, then things get messy. Probably some combination of the regulator/government/other brokers end up footing the bill.
Can a Hedge Fund (well, any big player) bust through their margin and
not be able to pay up? (Or is that conceptually impossible for some
reason I don't understand?)
It's the broker's (and the hedge fund's) job to make sure they don't (seeing as they're the ones who'll be liable). I imagine, for massive players, they'll give you more leeway than some tiny retail account. But ultimately if prices move violently and unexpectedly, it could happen.
But even if it does happen. Chances are if you were running a massive hedge fund, you got that way by being very good at what you do. So there are probably outside players who would be willing to bail you out in return for a stake in your fund/company.
A great example would be Knight Capital Group. 10 years ago, they were the largest trader for US equities. Through a series of unfortunate events related to a software update, their algorithmic trading software went haywire one morning. Over the course of 45 minutes of trading on August 1 2012 they lost $440 Million, which was enough to virtually bankrupt the company.
They raised $400 Million from a group of investors to plug the hole, and ended up being acquired just 4 months later.