Former Manchester United defender Stam took over from Giovanni van Bronckhorst during the close season but was only able to guide Feyenoord to three wins in its opening 11 league games.
Defeat to its bitter rival in Amsterdam left it languishing in 12th position.
Stam – who has also coached Reading and PEC Zwolle – told the club's official website: "I've thought about it a lot and for a long time. My final conclusion is that it is better for the club, the players and myself if I step aside."
Feyenoord technical director Sjaak Troost said: "The coach certainly still had our support and the players also wanted to continue with him, but if someone says, 'I no longer believe in it and don't want to carry on,' we respect that and it will stop. What a shame."