To exit Doom 64 map 24, No Escape:

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

On this map there are one or more Cyberdemon monsters,
depending on the Skill level that you're playing at.
To exit this map, you must destroy all of the Cyberdemons.
When the last one is destroyed, you exit automatically.

On all Skill levels, at least one Cyberdemon is trapped
in a cave-like area, behind twin sets of bars.  Here's how
to remove those bars:

Look for the long red wall that's on the far side of a
stream of acid.  There are 2 openings in this wall, go into
the left one, and look for the teleporter in there, around
the corner to the right.
Step on the teleporter, and you'll be sent to an indoor
area, and you'll be near the bottom of a staircase.
Go up the stairs, and follow the left-hand wall until you
come around to a balcony.  Go to the right side of the
balcony, and turn so you can see a platform just below the
balcony and to the right.
Jump off the balcony on to that platform (you may need to
use your Run button), and Activate the switch you'll find
there.
Now jump down into the main courtyard, and again go back to
the long red wall, jump down to the slime and go into that
opening that leads to the teleporter.
Step on the teleporter again and you'll beam back to that
area at the bottom of the stairs.  Go up the stairs and
follow the right-hand wall around to a new opening ahead on
the right.
Go through the new doorway to the room beyond, and Activate
the switch you find there.  This removes the bars that
blocks the cave-like area containing the Cyberdemon(s).
On Skill levels 3 and 4, there will be 2 Cyberdemons there.

Note that on Skill levels 1 and 2, there will be a Mega
Sphere located in the Cyberdemon cave.

When the last Cyberdemon is destroyed, you exit this level
automatically.