To exit Doom II level 3, The Gantlet:

There is a very large, rectangular area dominating the east
side of the map.  Go to the far east side of this area, and
you'll find the blue key on the ground, close to a pair of
blue-keyed doors.
Grab the key and open the left-hand door.  Follow the hall
north and west, to a switch you'll see ahead of you, and
Activate it.

Now head back out into the large rectangular area, and go
to the top of the western-most staircase.  From here, climb
up on the northern ledge that runs west-to-east alongside
the stairs.  Follow this ledge east to a narrow bridge on
your left, which leads north to a teleporter booth in the
north wall.

Step into the teleporter, and you'll re-appear in a small
evelated room to the west, and you'll be facing southeast.
Open the door to the south and go through.
Drop down into the courtyard and grab the red key you'll
find there.

Turn right and go to the pair of switches on the west wall.
Activate the right-hand switch, and immediately turn left
(south).  That switch causes a lift to drop down in the
southwest corner of the courtyard.  Go step on the lift and
ride up.
At the top, follow the ledge to the red-keyed door.  Open
it to reveal the exit switch in the room beyond.

PC Doom II players can click here to access a demo showing
how to exit this level.