To get the chain saw on level E1M2, Nuclear Plant:

(Note: For a video demonstration of the walkthrough below,
click here.)

To the northwest of the level's starting point is a door
that you'll need a red keycard to get through.
Head west through this door, and take a right, down a short
curved passage that leads to a small slime-floored room.
Look for a pillar in the center of this room and Activate
the switch on its back side.
Leave this slime-floored room, and as you head back towards
the red-keyed door, you'll see a newly-opened wall section
ahead of you, located just south of the red-keyed door.
Walk into this new passage, down a few steps, and turn
right.
Walk a couple paces forward and you'll see a wall edge, or
partition, up ahead of you and a little to the left.
To the right of this partition is a single passage that
continues off to the right: Ignore that path for now.
Instead, walk to the first opening just to the left of that
partition, and continue on in that direction, staying close
to the right-hand wall.
Continue forward (west) and up ahead you'll see a barrel in
the corner.  Walk up to this barrel and turn left (south).
You'll see a green armor jacket ahead of you on the floor
against the wall.  The wall panel right next to the armor
is a secret door.
In SNES Doom, this panel can be opened like any regular
door.  In all other Doom games, use your fist, pistol,
shotgun or chaingun to attack the panel, and it will open.
Enter this newly opened passage and follow the winding
stairway up to a windowed area where you'll see the chain
saw on a pedestal.  A switch in the corner of this room
will cause the pedestal to lower, allowing you access to
the chain saw.

PC Doom players can access a demo of this level that shows
how to get the chain saw, by clicking here.