Which institutions are in the eurodollar market? My main question is: how much of the eurodollar market is claims issued by depository institutions or Bank Holding Companies?

It's about a $4tn market, but who's in it? Who is funding themselves in eurodollars and who are the dominant players?

I'm trying to figure how much of the market is outside the banking system. Any links/research is greatly appreciated.
Eurodollars are, by definition, held (on the liability side) inside the banking system. They're US dollar deposits held either outside the US or inside the US via an International Banking Facility. On the asset side, eurodollars are typically lent by nonbank financial institutions. This means that the eurodollar market links the two segments of financial markets: it's typically nonbanks providing short-term funding to banks.

For an introduction to nonbank financial intermediation, I'd recommend "Shadow Banking" (Pozsar, et. al., 2010).