How to access the keys and the exit in PC Doom II map 24,
The Chasm:

When you first enter this level, you'll appear on a ledge
that's near the middle of a gigantic room that dominates
the east half of the map.

Just to the north of your starting position (towards the
top of the Automap screen), there is a tall, narrow
building at the end of a high, curving walkway.  Inside
that building is a small maze of green slime-covered walls.
The blue key is in the northwest corner of that maze.

After you pick up the blue key, step out of this building,
and head south back along that high, curving walkway.  The
south end of this walkway joins the ledge where you first
appeared.

At this point you can go 2 directions: left (southeast),
and right (west).  Go right to follow the ledge west, then
north to a blue-keyed door, and go through.

Ahead of you to the west is a very narrow, high walkway
above another sea of slime.

Walk forward/west to the first intersection, and go right
(north).  Follow the walkway as it bends left/west for a
couple of paces, then take the first right and go north.

Ahead you'll see a large 4-sided green structure hanging
down from the ceiling.  Behind that structure, a little
further to the north, is a smaller red-brick structure
hanging from the ceiling.  Just beneath the red structure
is a small square platform.

Go step on that platform, and you'll be transported to a
smaller slime-floored area at the north edge of the map.

When you materialize you'll be standing on a small platform
and you'll be facing east.

Warning: Do not fall into the slime in this room.

Off to your left (north) there will be a platform against
the wall.  You'll have to get to this platform by carefully
following another winding, narrow walkway.

Once you get to the platform, you'll see 2 teleporter pads:
A red one to the left, and a white one to the right.  Step
on the white/east-side one.

You'll appear in a smaller room at the southwest corner of
the map, and you'll be facing north.

This room has 4 sections, and each section has a platform,
and a switch.  Each switch controls a specific platform in
the room: Lowering it temporarily for a couple of seconds.

The red key is on a platform at the north end of the room.
To get to that platform, Activate the 4 switches in the
following order: East, West, North (to access the red key),
then use the South switch to access a teleporter that
allows escape from this room.

Once you teleport out, you'll appear back on the white
teleporter in that smaller slime-floored area at the north
edge of the map.

From here, carefully follow the narrow winding walkways to
another platform at the northeast end of the room, where
you'll find a red-keyed door.

Open this door to access the Exit door; Open that and step
through to finish.

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