In some board games, it is possible for a player to be in a situation where there are no legal moves to be made, such as stalemate in chess. In contemporary chess, a stalemated game is a draw; historically there was no standard rule, and stalemate was sometimes considered a loss for the stalemated player — or sometimes a win (like in Chinese Chess). Chess also has a rule that the game is drawn if no possible sequence of moves from the current position can lead to a win. The most obvious example is when both players have only their king left, but there are other possibilities, such as the 16 pawns forming a complete blockade, that are covered by the rule even though they will never arise in realistic play.
The 50-move rule (the game is a draw if no pawns have been moved and no captures have been made after each player has taken 50 consecutive moves at any stage of the game) was added because it was thought that the game was Unwinnable when it had devolved into such a state. Then someone found a way to mate a player this way...
Another specific case is the "threefold repetition" rule, whereby a game is drawn if an identical position occurs three times in the same game, implying that no progress is being made.
Empire Builder has the potential to create this for one or more players and make things very annoying for the others. If a player is very low on cash, a natural disaster like a flood can destroy enough of their railroad that they can't afford to repair it and are cutoff from any city where they could make more money. As a last resort they discard their contract cards and draw a new set of contract cards hoping to get one that will get them the money to proceed. However, this increases the chance that another disaster card will be drawn which only makes things worse. They have legal actions in the game but those actions get them nowhere. The other players now have to deal with the possibility of a natural disaster card every other round instead of every five to six rounds. Also having one player sit around for another hour or two locked in an unwinnable situation is not a pleasant experience for everyone. Since lending other players money is not allowed, the others players will find a way to pay the stuck player rent money for using their railroad just to get them back into the game.
In the board game Hero Quest, it is entirely possible to lock the game into an unwinnable state by making either the Elf or the Wizard use the spell "Pass Through Rock" then passing through one of the many boulders that are used specifically to stop you from going to rooms to have no way in and nothing of interest thus trapping you on one side of the board with no way out.
An Obvious Rule Patch changed it so that if you "Pass Through Rock" and end your turn in one of the empty rooms or hallways, you are considered to have ended your turn inside solid rock, and died messily.
In some Hero Quest games, the rules state that the players can only search once in a given room. This can lead to some of the quests becoming unwinnable.
Betrayal at House on the Hill boasts fifty Haunt scenarios that are randomly chosen each time you play. However, due to the random nature of the game, it's sometimes possible to end up in a situation where one side literally has no chance of winning. For instance, the Traitor becomes a near-invincible monster with one weakness... only by sheer chance, they happened to find that item and were carrying it when the Haunt started. Leaving the heroes with no way to retrieve it. To make matters worse, some of the scenarios as originally published had conflicting or unclear rules, which could also render a scenario Unwinnable. However, since the nature of this game is not very competitive, in most such situations reasonable players will elect to veto the haunt in favor of something more fun.
It says something this game ended up getting a 20-page errata book to correct all the errors, and some are still there.
In-universe example: on the SitcomNewhart, George invents a wildly popular board game where every space landed on gives the player 3 points. The first to score exactly one million points wins. Not until someone actually reaches that point does anyone involved realize that one million is not divisible by three.
The board game Dragon's Gold normally revolves around players negotiating over which items each will take from a dragon's hoard. However, the game features an alternative mode where the loot is divided according to fixed rules. In this variant, if all a player's forces are committed to taking down a dragon whose hoard contains a cursed item (although possibly a large amount of treasure as well) but due to interference from other players cannot beat the dragon, the player is effectively out of the game. Without any soldiers they cannot attack other dragons, nor can they pull back their existing ones. Normally, they could ask another player for help, but under the fixed rules for loot distribution it is impossible for them to promise that player that they will not end up with the cursed item, meaning that no player will accept their offer.