Virtually every government in the world is borrowing heavily in order to stimulate the economy. Who's lending?
Governments have been borrowing heavily for centuries, so to some extent there's nothing new here. (See here for the CIA's ranking of nations by government debt as a percentage of GDP.) It also isn't necessarily true that every government is borrowing heavily to spend on stimulus: a lot of governments are spending only a tiny fraction of their GDPs on such packages, and many governments aren't spending anything at all. Still, there would seem to be something intuitively weird about a situation in which lots of governments are increasing their borrowing at once. How does that make sense?
One answer is that we're only talking about government borrowing, not total borrowing. Even if every government in the world wants to borrow more -- I'm not sure that's the case, but it could be -- the money can come from private lenders. (A lot of US government debt is held by its own citizens.) The second answer is that an increase in government borrowing doesn't sap some finite and predetermined pool of lending. The increased demand for funds just increases the cost of borrowing -- that is, it'll increase long-term interest rates.
(Of course, this is a problem for two reasons. First, the increase in interest rates reflects an increase in risk. The more a country increases its debt, the greater the chance of default. To off-set the additional risk, the borrower must offer a higher rate of return. The fear here is that at some point the risk of default becomes so great that demand will drop off entirely. The increase in interest rates also creates a second problem: the higher rate of return on government debt will crowd out private investment. Why would you buy the debt of a corporation if you can get a higher rate of return from the government?)
So anyway, who is buying up the governments' debt? Lots of people. The US (to take the example I know the most about) has about $11 trillion in debt. About $4.3 trillion of that is "intragovernmental": debt held by other US government institutions like the social security trust fund. The rest is the "debt held by the public." Most of this is in the hands of foreign central banks, the largest of which is China. Some of it is held by pensions and mutual funds and other investors. You can find the full breakdown here.
China and others are buying US debt because, despite everything, the US is seen as relatively safe investment in uncertain times. (China also likes US debt because it helps keep Chinese exports cheap.) Countries that are a bit riskier will have that reflected in a sovereign debt rating and will have to offer higher yields. But they will still have buyers at those higher yields -- either private institutions or governments that are still net creditors.