They can be opened by possessing a number of cards greater than or equal to the number on the gate and pressing "C" while standing in front of it.
The 08 gate is on the western part of the Beach area. It leads to another HP upgrade.
The 16 gate is on the eastern edge of Young Town. It leads to another HP upgrade.
Located within the Underground Labyrinth. It leads to a HP upgrade.
Located within the Terminal.
Located in BLANK.
Located within the Nexus.
"The 50 card door is an ending to Young's story, at least within the game's narrative. It will never open."
This gate never opens, even if you hack the game to say that you have 50 cards. But even if you further hack the game to cause the gate to open (or use Cheat Codes to get past it), there is nothing past the gate but a blank expanse of nothingness (see map of BLANK.). This gate's purpose is not to obstruct your progress, but to be symbolic of an impossible task.
Sage changes the number on the 36 Gate to 92 in an attempt to discourage Young from passing through. However, the 92 Gate still opens with only 36 cards. The number 92 is a lie.