Is there any way to escape Ana's tranquilizer dart?
There are three ways to remove the effects of the dart or escape.
You get stunned by the Dart, but then you get damaged. This is the normal effect of the dart.
You dodge the dart in any way possible.
The dart's effect wears off in 5.5 seconds.
Most likely your example is of a person being shot by a dart, seeing the animation/hearing the sound and quickly going into Cryostasis before the dart actually hits you. As Overwatch is a prediction game, the game predicts that the dart will hit you, and if you manage to react in the 20ms before the prediction comes true, then you're safe and healing and the dart never hits you.
Once you're hit by the dart, there is absolutely nothing you could do. You can't use your skills, you cant move, you can't jump, you can't ult - Nothing. Only damage will save you, or if the time of the dart runs out.
I personally like to go against an Ana, try to predict when would be a good time for her to shoot me, and do an unexpected move (like fly up in the sky with Phara). Most of the time Ana misses after firing and I can carry on fighting.
The sleep from a Tranq Dart only ends after 5.5 seconds or after taking damage from the enemy. The sleeping hero can't act at all, and his teammates can't do anything to wake them up.
You can dodge it or block it with Reinhardt's shield. If you get hit, then the only ways to get out of the stun effect are to get hit by the enemy or wait the 5.5 seconds until it wears off.