Endurance can be thought of as one's ability to drive for extended periods of time as well as overcome obstacles. The first part should be obvious. To convince yourself of the second part, consider sprint and endurance racing: a mechanical failure in a sprint race often means the team ends the race early, but in an endurance race the mechanics are expected to fix it and the driver to continue racing until the end.
The typical session length of 20 to 30 minutes is the period of time that the typical driver can stay focused on driving and not get too tired, mentally and physically. Staying focused for an hour straight is not going to magically happen - you need to work on it. Running true open track days, where you can drive for as long as you have fuel in the car, is a great way to practice driving long stints. You can also pace yourself to run as long or as short stints as you like. As a bonus, the enormous amount of track time you can get will alone improve your skills substantially, especially on the track in question.
Instructors often run consecutive sessions between instructing and driving themselves, and as such they benefit from having practice driving long stints before they start instructing as well as continue improving their endurance while they are instructing, if they use their allotted driving time.
In the context of HPDE, overcoming obstacles most commonly refers to overdriving. I detailed my strategy for dealing with overdriving on that page. Overdriving can be converted into endurance if, after analyzing an incident, you return to the track and finish the session, possibly at a reduced pace. Especially for spins and four-offs, driving the same track segment at speed can be daunting. You will only overcome the fear by slowly getting your confidence back - by driving.
If you keep analyzing your mistakes and returning to the track, you will be improving your recovery skills. With time you will be able to perform overdriving analysis while driving - perhaps to the pits if the offence requires a consultation with an official, perhaps simply while driving if you "saved" the situation. You can get to the point where a couple corners is all it takes to recover from an overdriving situation that may have sent a novice driver into the pits.
Endurance can be your friend in competition - in time trials, your best lap time may well happen on a lap where you put two wheels off track because you took a corner that much faster than usual. If you can recover from the two off without slowing down much, you turn an overdriving situation into an advantage. (This actually happened to me. At one event I had a transmission issue and wanted to drive the car as little as possible. I only ran one session but made every lap count. In that one session I went two off three times in three different corners, but posted a lap time I was happy with.)