Not directly accountable but still much more accountable than Facebook is. Even in a non-elected position there is still a duty to the people and greater good rather than fiduciary duty to shareholders.
The only real way the Fed is accountable to the people and greater good is to the extent that they don't screw the people so obviously and badly that the people (including the police and military) revolt against them, which is the same level of accountability for any dictatorship