Can the stock market overall go up if 'big money' isn't buying stocks?
Big money is institutional & professional investors, aka "smart money." Mutual funds, pension funds, banks and other big financial institutions including hedge funds, large private investors ("family companies"), and their professional managers & traders.
Some of them are always in the stock market as well as the alternatives markets, bonds, & cash markets. They are so large and dominant that to a significant degree they "are" the market -- it does not exist without them.
In general, "out" of the stock market means all of the above.
In storms, the assumption is cash -- money market accounts, cash equivalents, US Treasury Bills (massive, and I mean truly massive, amounts of money went in this category in 2008-9) and equivalents for highly regarded national governments (UK, Japan, Euro deposits...), time deposits including CD's and commercial paper (a "corrupted" asset in this last crisis), and money market assets (suspect until guarantee by the Fed reestablished confidence).
yes, they often are in cash or CDs waiting to invest their money. They are usually under pressure to try to invest it in stocks because they need to make money for shareholders. Try HOGS if you want to make something.
Pension plans, charitable (especially university) endowments, hedge funds.
They go into government bonds. CDs don't work because banks aren't big enough to absorb this kind of money and it would be way over the FDIC insurance limit.
The stock market can't go up based on individual investors because they just don't have enough money compared to the institutions.