On this airport, it seems the runway end with a wall followed by a mountains. Thus the pilots physically can't overrun the runway and if they decide to interrupt the landing, the go-around maneuver seems to include a very tight turn to avoid the mountain unless performed very soon during the approach. Is it possible to perform a go-around, and a which point it becomes impossible for this particular airport?
No, it is almost impossible.
The only runway (06/24) Lukla Airport is just 1500 feet long. This article lists some interesting facts:
The topography of the place makes any go-around impossible. At the south, the runway is the end of an angled drop of about 2000 feet. This cliff is fenced off as a precautionary measure. At the northern end of the runway there is a huge mountain terrain. It leaves no room for error.
A pilot mentions the same here:
"Because there is no way to go around again, we have to calculate many things like air speed, tail wind, fog. If you don't do the proper calculation or proper exercise, then it (an accident) happens."
Planes land on runway 06 (Heading 060 or ENE) and takeoff from runway 24 (Heading 240 or WSW). This picture shows the cliff on North:
There have been several accidents at this airport.