I use the following houserules:
1 - Exhaustion
When a player gains a level of exhaustion, he/she chooses one of his stats and suffers a disadvantage penality to all related rolls (skills, saving throws, attacks...). If all her/his stats are affected, he dies.
I was reluctant to give players levels of exhaustion with the original table, because I felt like the first penalty litteraly ruined the game for the player affected. Disadvantage to all ability checks is way too harsh. If you have one player with a level of exhaustion in the groupe, he/she has to leave everythings that's not monster bashing to the other players and is probably not having a good time.
I felt like this solutions is a bit more interesting, realistic, and leaves a choice to the player. A seasoned warrior suffering from exhaustion might start by thinking less, or get a bit grumpy... but will still be able to lift the heavy stuff. Whereas a tired wizard might first become physically weak.
With this rule, I have less hesitation to give my players levels of exhaustion. Wich brings me to the second house rule
2 - Death and Exhaustion
To avoid players healing a few hp to a downed friend only to keep him up for a round, and not caring wether he/she gets knocked out again, when a player gets to 0 hp, and comes back to consciousness, he/she gains two levels of exhaustion.
This way, getting knocked out feels a little bit more dangerous, since you need a long rest to recover from a level of exhaustion. Three downs means death. And if you get knocked out two times, you better rest instead of going adventuring the day after.
But even with four level of exhaustions, the player can at least keep his/her main stats working... though admittedly, he/she won't be any good at anything else.
3 - Dungeons and rest
When players get into a dungeon situation. A closed building or underground caves, with lots of danger, I use the quicker rest rules from the DM's Guide : 5 minutes short rests, and 1 hour long rests. But at a price. When they get out of the place, and the tension drops, exhaustion catches up to them. They get one level of exhaustion for each short rest they took after the first, and two levels for each long rest. They can't accumulate more than 5 levels this way, or by being knocked down. If a player tries to rest in this state, he might simply fail gain any benefits, or collapse for a few days.
I use this rule to prevent dungeon camping. It doesn't feel right to have players stop for 8hours in a den full of monsters. I felt like this rules adds tension, players are less resting than mustering all their inner rescources to keep going.