They hold most 30 year mortgages. Subprime mortgagages, ARMs etc were securitized. If the problem was Freddie or Fannie not getting paid it would be a different problem.

Lehman Bros went bankrupt, as would many other financial institutiosn without a bailout because they held a lot of garbage paper (including subprime mortgages) and were very illiquid. Many European and Asian banks had huge US mortgage exposure.

The problem was extreme mispricing of risk and illiquidity.

At the time of the crisis they were listed on the stock market and then put into 'conservatorship'.

They bought mortgages and resold them on as mortgaged-backed securities, presumably to a mix of US and foreign entities.