To access the Secret areas on Doom 64 map 24, No Escape:

There are 3 Secret areas on this map which can affect a
player's Secrets score: One area contains a BFG 9000,
another contains a backpack, and a third contains a plasma
rifle and some ammo.

THE BFG 9000
------------
When you first appear in the level, Activate the switch in
front of you, then walk forward past the switch, and drop
down into the courtyard.
Before doing anything, take note of the long, red wall
ahead of you, running along the far side of a toxic river.
Down along the bottom of this wall are 2 visible openings:
One is a cubbyhole where a Radiation Shield can be picked
up; And around the corner to the left of that opening is
another doorway, leading to a short curving tunnel where a
teleporter can be found.
A hidden third door exists a few paces to the left of the
other two doors, and can be opened temporarily to reveal a
secret chamber where the BFG 9000 is located.
At the far left end of the long red wall is a large doorway
which is blocked off by vertical bars.  Along the left and
right sides of this door are some tall, 4-sided pillars.
Each pillar is made of a stack of 7 cubes, and each side of
each cube has a face on it.  But on the left-side pillar,
the fourth cube up from the ground has a face that doesn't
quite match the others.
Now, from the place where you first dropped down into the
courtyard, turn left and follow the left-hand wall a few
paces, until you come to a stone pillar/elevator.  Activate
it and it drops; step on and ride up.
Get your pistol (or other bullet/pellet weapon), and shoot
that 4th cube on the left-side pillar.  This causes the
hidden third doorway to open along the bottom edge of the
long red wall.
It will remain open for a few seconds, then close again.
Run to it to get inside, where the BFG is found.
Step out and head for the opening to the teleporter.

THE BACKPACK
------------
On one corner of the main courtyard, you'll find a wide set
of steps leading up to a large square room that contains 16
tall pillars.
One of these 4-sided pillars has three switches on it.
Walk around to the side of this pillar that has no switch.
Activate this side of the pillar, and another pillar near
the center of the room will sink down into the floor.  On
top of that pillar is a backpack that you can now grab.

THE PLASMA RIFLE
----------------
To get the plasma rifle, you first have to remove a set of
3 vertical bars which block access to a doorway that leads
to a long curving stairway.
Look for this barred doorway along one edge of the main
courtyard.  The doorway is between a pair of gargoyle-like
statues that sit upon tall, 4-sided pedestals.
When you're facing the barred doorway, turn around 180
degrees to look behind you, towards the structure that
dominates the center of the couryard.
Walk up to the raised pillar/lift that's against this side
of the central structure, and Activate it.  When the lift
drops, ride it up and step into the tiny recessed room at
the top.  Activate the switch there to remove the 3 bars
that block access to the spiral stairs.
Jump back down and go through that newly-opened doorway.
Walk down the spiral steps until you are on the 7th step
down from the top.
Turn right and search the wall for a secret panel.  This
panel opens to reveal a small chamber containing the plasma
rifle and some ammo.