Acquire the red key in Doom 64 map 29, Outpost Omega:

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

When you first appear in the level, you'll be facing down a
hallway, with a small rectangular room to your left, and
another to your right.  Ahead of you in the distance is a
sealed door.

Turn right, enter the small rectangular room and activate
the switch you'll find there.  This opens the sealed door
at the end of the hall.

Walk up to that now-open door but don't step through yet.
You should see a large room through the doorway.  In the
middle of the far wall of that room is a hidden panel, and
behind it is the red key in a tiny chamber.

To your left and right, just in front of the open doorway,
is a panel on each wall.  The left one has blinking lights,
the right one does not.  Activate the right-hand panel like
a switch, and the hidden chamber in the distance will open
momentarily.  Quickly run to grab the key before it closes.

Note that if you pick up the rocket launcher in the middle
of the room, monsters will appear in the cages around you,
and the open doorway will close again.  Killing all of the
monsters will cause that door to reopen.

If you did not reach the key in time, simply go back to the
panel by the open doorway and try again.