I need to replace a computer brain box for my 2001 Saturn L series Wagon. I'm thinking it is located behind the glove box on the passenger side but how do I get to it?

Let a shop do it for you for the serial number of the vehicle has to be burned into the replacement computer for you

The ecm is buried under the passenger side windshield cowling. The bcm sits on top of the glove box. Your 4cyl engine uses a timing chain that may have suffered from lack of lubrication due to poor oiling controls. Although you're describing a symptom, ecm's rarely fail ($$$), as your description may have everything to do with the bcm and the factory security system; starting, running then stopping. Have you reviewed your owner's manual for resetting the security system?

Jumping on ecm replacement won't be as easy as you may think as there's a factory security feature that forces all the system computers to communicate with the body control module (computer) before allowing the new kid to work. There's a relearn routine that's part of replacement procedures and unless your ecm find gives you this information, you'll be in more trouble than you can imagine.