- Is the research reliable?
Well, the Monthly Notices of the Royal Astronomy Society is one of the longest running astronomy journals in the world, and, to my knowledge, has never done anything substantial to impugn its reputation. It also has a comparatively large impact factor. All signs that the peer review is considered of good quality.
- How can such a thing be stable? Is there any particular process that keeps one star inside the other?
Why wouldn't it be stable? More gravity means more fusion, not less.
The theory says it's a companion star that goes nova, and then is gradually de-orbitted into the larger gas giant.
- What even /is/ such a body? If you were to travel from the outside to the midpoint of the body, would you encounter two barriers of destructing heat, with some emptiness (I'd like to say "vacuum" but of course space is not exactly a vacuum) in between?
Or is it actually just something entirely unlike what you would imagine when someone says "star within a star"?
Oh, and just now I realize you hadn't read the summary. It's a neutron star inside a star. A neutron star is essentially a block of neutronium(essentially a gigantic neutron only nucleus) with some attached hanger on high energy plasma around.