When I introduce the GDP figures to my macroeconomics students, I point out that the federal government accounts for a much smaller portion of the economy than many believe. Federal government purchases - i.e., the federal part of G in the national income accounts - was 8.3% of GDP in 2010, and more than two-thirds of that was military spending. The federal nondefense part - all the stuff that we think of as "the federal government" like the FBI, NASA, the State Dept. and so on - was a mere 2.7% of GDP.
One reason that people think that the federal government is larger is that federal spending was $3.7 trillion last year, which would be 26.5% of GDP. But most of that is transfer payments - money sent to people (and, to a lesser extent, state governments). Mainly, this is social security and medicare. In the national income accounts, most transfers end up in the "C" (consumption) component of GDP when the recipients spend the money.
The chart below, from BEA data, illustrates that nondefense federal government purchases are only about 10% of federal spending.
So when zealous congressional republicans say that they are going to cut our "big government" by $100 billion*, the consequences are quite severe because the cuts almost entirely come out of that small slice. This analysis by the Center on Budget and Policy Priorities provides some of the bloody details, including cuts of 12.7% in Labor, Health and Human Services and Education, 14.5% in Interior and Environment and 26.1% in Transportation and Housing and Urban Development.
*It depends on how you measure it; as the CBPP analysis explains, the "$100 billion" figure is relative to the president's proposal and is for an entire fiscal year (which runs from October through September), while the cuts are for the remainder of fiscal 2011.
The CBPP analysis comes to my attention via the invaluable Ezra Klein.
Update: Paul Krugman has more.
Update #2: So does Bruce Bartlett.