According to the Securities Industry and Financial Markets Association (SIFMA), as of Q1 2017, the U.S. bond market size is (in billions):

Note that the total federal government debts recognized by SIFMA are significantly less than the total bills, notes and bonds issued by the U.S. Treasury Department, of some $19.8 trillion at the time. This figure is likely to have excluded the inter-governmental debts such as those held by the Federal Reserve and the Social Security Trust Fund.

History
Bond investments
Participants
US bond market size