The coordinates for this episode point to a white dwarf named GD 394 located at a distance of 164 light years (50.3 pc) in the constellation Cygnus [1, 2]. A white dwarf is the core left behind when a star about the size of the Sun dies. The Sun and other stars like it are currently at the stage where they are fusing hydrogen into helium in their cores. However, as a Sun-like star gets older, its core will fill up with helium, and the Sun-like star will go through a series of changes including growing to become a red giant. Later on, the star will fuse helium into carbon or oxygen in its core but then will not have the mass to trigger the fusion of carbon or oxygen into heavier elements. Once a core of carbon and/or oxygen builds up, the core does nothing, but fusion still continues around the core, and eventually the outer layers of the star get blown away, leaving the inert core. That core of carbon and oxygen, which may retain some sort of thin layer of hydrogen and helium, becomes a white dwarf.
The universe contains a lot of white dwarfs, and if GD 394 was a boring white dwarf, I wouldn't have created an entire episode about it, so it has to be weird in some way. The weirdness started in the 1980s, when the International Ultraviolet Explorer, which was one of the first ultraviolet space telescopes, detected the presence of a substantial amount of silicon in GD 394's outer gas layers [3]. Silicon is an element much heavier than carbon and oxygen that could not have been created by fusion in the core of GD 394 before it became a white dwarf, so the white dwarf shouldn't contain that much silicon if any at all. Later on, people also found extra-large amounts of aluminum, phosphorus, and iron in the outer layers of GD 394 [4, 5]. Again, these are elements much larger than carbon and oxygen that could not have been formed in the star's core before it became a white dwarf, so it was not immediately clear where these heavy elements came from.
What got weirder is that, when people started studying GD 394 more carefully in the 1990s with another ultraviolet space telescope named the Extreme Ultraviolet Explorer, they discovered that the ultraviolet brightness of GD 394 varied on a period of 1.150 days [6]. Confusingly, though, people doing follow-up observations in the 2010s did not find any variability in the ultraviolet light [5], but then they did find variations in the light at visible wavelengths [7]. They also came up with the slightly more accurate measurement of 1.146 days for the brightness variation period [7].
Anyhow, this now indicated either that something was orbiting GD 394 very closely or that something was happening on the surface of the white dwarf. This includes a whole bunch of various hypotheses, a lot of which involve exoplanets or something like that. One hypothesis describes material containing a lot of heavy elements falling onto GD 394 in just one place, forming a "dark spot" (or actually a slightly less bright spot) on the outer layers of the white dwarf. (It wasn't exactly clear to me how infalling material would just land in one spot, but I did not create the hypothesis. Maybe it has something to do with magnetic fields.) Anyway, another hypothesis is that a planet is orbiting very close to GD 394 and is evaporating, forming a gas cloud that contains a bunch of heavy elements that partially obscure the light from the white dwarf every time the planet passes between us and the star [5]. A third hypothesis is that a bright spot on the surface of GD 394 has been created by magnetic fields connecting the star to some sort of orbiting planetary object made of iron [7].
GD 394 seems to be a real mystery, and so I would expect astronomers to continue to try to observe the star to understand exactly what's going on. Also, this could involve exoplanets, and that really excites astronomers and their sources of research funding, so that's another reason why astronomers will continue to observe GD 394.