At the end of a 4 player game, 3 of the players had fewer trains than spaces required to complete the smallest trip (we each had 8 trains and you needed a minimum 9 trains to complete the smallest trip available). Only one player could continue to select train cards to reach a destination. What happens in that scenario. Does that player continue picking destination cards until he finds one where he could complete the trip or is the game over?
It sounds as though you have misunderstood how playing train cards to claim routes works.
On your turn, you can either draw train cards, draw destination tickets, or "claim a route". A "route" is the spaces between any 2 adjacent cities; routes are from 1 to 6 trains long. You can claim any route that is still available if you have the correct train cards to pay for it; you don't have to have a matching destination ticket or anything like that.
Importantly, you can only claim 1 route at a time, so for example you would not be able to claim Los Angeles to New York on a single turn, even if you have that destination ticket and enough train cards in your hand to play all of those trains. Instead, a single turn would be for example playing 2 green cards to claim New York to Pittsburgh, and then scoring 2 points because that's what a 2-length route is worth.
So in your example, someone with 8 train cars will always have a route they can claim. Even if they can't finish their destination ticket with only 8 trains, they can still play those trains on the board for points, 1 route at a time.
From the rules:
The object of the game is to score the highest number of total points. Points can be scored by:
Claiming a Route between two adjacent cities on the map;
A player may claim any open route on the board. He is never required to connect to any of his previously played routes. A player may only claim a maximum of one route, hence connect two adjacent cities, never more, on his turn.
First, it's important to clear up the difference between a route and a destination. A route is just a leg connecting any two cities, whereas a destination card is one of the hidden cards players have. Even if players can't complete any of their remaining destination cards, they can still claim routes for points, and help end the game. While they will lose points for any destinations they haven't finished, they may want to end the game sooner to prevent other players from finishing their destinations. It from your question like players can still play routes even though they can't finish destinations, in which case there are still things all players can do.
That said, let's take your question at face value and examine what happens if a player has train cars left but not enough train cars to finish any available routes.
As was answered in At the end of the game, can you play a route that is longer than your number of trains? you can't play a route longer than your remaining number of trains. So, if a player has more than two train cars left, and the only remaining routes are longer than that player's number of train cars, there is nothing that player can do except for draw train cards and destination cards. Furthermore, if you did end up in a situation where only one player had enough train cars to claim the remaining routes, then it would be up to them to claim a route long enough to end the game.
However, the maps should be large enough to prevent that from typically happening. For example, there is one map currently with a 9 car route (the situation posed in your question), which is Ticket to Ride Nordic Countries. But it has only a single 9 car route: Murmansk to Lieksa. Ignoring this route, there are 188 other spots for train cars and only 120 train cars come with the game. In fact, there are over 120 spots on of routes that are 3 or fewer in length, which a player will always be able to claim (since if a player has fewer than 3 cars, the game will be over).
It is theoretically possible on Ticket to Ride USA, which has a in the balpark of 130 -140 train car spots on routes of length 3 or shorter, and there are 240 train cars in the game. Getting to such a situation would require players to focus heavily on acquiring all the shortest routes first, which is a bad plan because longer routes have much higher points yields, but it is possible.