Unwinnable by Mistake/Video Games/Final Fantasy

Final Fantasy VI

In the World of Balance, there's a tiny patch of land to the right of Nikeah just big enough to land the airship on. But if you land there and go into the town, then there's no way to get back--the town blocks you from reaching that patch on the World Map, and all of the exit points within the town spawn you to the left side. There's nothing preventing the player from saving the game in this situation. And without the airship, the game is unwinnable. This example is especially sinister when you consider that the primary reason a player is likely to take the airship to Nikeah is to pick up Mog's Water Rondo dance, which is Lost Forever after a brief window of game events.
The original Japanese version had a bug in the World of Ruin that would send you back to the World of Balance. You can't come back to the World of Ruin. This was fixed in all subsequent releases.

There's a point where the characters have to stop a missile launch, but to get the codes you have to help out some enemy soldiers. If you refuse to help them, then you cannot change the missile launch, and it's possible to save after refusing to help, which means you'd have to start the whole game over if you don't have a back-up file.
After having access to chocobo, it is possible to ride a chocobo through shallow water onto a barren island in world map, get off, and get stuck. Consider that the world map is the place where saving anywhere is possible, it is entirely possible for a player to save here and essentially get stuck for eternity.

In Final Fantasy X, there is an airship battle against "Overdrive Sin" near the end of the story. Meaning that, basically, you can only inflict damage with magic because of how far away the target is at the start of the fight (the only one who can hit it with melee the entire time is Wakka). Sin's Overdrive gauge will fill up a bit every turn, and once it's full, it'll use an attack that causes an automatic game over. Doesn't even inflict damage, because it's plot. If you have only one file, you save right before the battle, and you're underleveled or have been neglecting to train Wakka or your magic users (primarily Lulu at that point), then you're completely screwed.
The siege of Bevelle immediately after fighting Evrae is also like this for some people if they're under-leveled, although having some equipment abilities can make it drastically easier (Stonestrike, Fireproof/Fire Eater).

Final Fantasy Adventure requires the player to know in advance that they need Mattocks and Keys. The need of Mattocks is eventually eliminated by the Morning star weapon; but if you don't have enough keys before entering Dime Tower, or if you save when you don't have enough keys and can't go back...then you can't beat the game.
- But that's not all. It is entirely possible to miss the Morning Star. You receive it as a reward from a boss battle; but the battle is optional, and getting to it requires traveling through a frustrating maze.
- It's possible to save in a room that you need a key to get out of, where no key-dropping enemies spawn.
- Note: Resetting the game causes all doors to close. If you have a key shortage, then even going back the way you came might be impossible.
It is possible to save at any time in this game through the menu instead of using save points. In at least one room of the airship around the middle of the game, it is possible to be permanently stuck if you save in that room then load that save, you will appear in a fenced off section of the room that's normally inaccessable with absolutely no way of getting out.

Final Fantasy Tactics has one where in Riovannes Castle right before the final battle with Wiegraf. The first part of the fight is 1v1 with Ramza, and if you don't have the right equipment and/or skills, it's basically impossible to win. If you save right before the fight, you have no chance to go back to level grind or purchase equipment, so if you don't have those skills and equipment, you're forced to restart the game.
- Owing to the games mechanics, you can still be screwed even if you have 20 levels on him.