4.5 US Quarts, including filter (you do change your filter every time you change your oil, right?) This is according to the owner's manual.
what is the oil capacity for 2004 honda odyssey
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2006 honda odyssey engine oil drain plug torque
all odyssey's take 4.8 quarts of oil
4.5 Quarts (with oil filter change)
4.5 quarts with oil/filter change.
5W20 Mobil
4.6 quarts with filter change
Honda owner's manual states that the oil capacity is 4.7 quarts.
According to http://en.wikibooks.org/wiki/Automobile_Repair/Honda/Odyssey/Change_engine_oil_and_filter, you need 5 quarts of oil.
The crankcase capacity for the 2005 Honda Odyssey is 5.3 quarts. The model comes with the Maintenance Minder System Indicator to help remind owners when the oil change and other maintenance chores are due.
Oil Change Time