I don't know that there is any natural explanation for this planet, even within the confines of Pseudoscience. Certainly it is possible in the realm of sci-fi technology to create such a planet.
there are two stars orbiting each other, and there is a little spot exactly in the middle of the two that is suitable for life.
This is probably possible in theory that a planet happens to have formed exactly at the Lagranian Point between the two stars. I find it extremely unlikely to happen.
TECHNOLOGY SOLUTION: As this is probably extremely unlikely to have occured naturally, it's more likely that someone (doesn't have to be your people) PUT the planet there
Because of the gravity of the stars and the rotation of the planet, everything lives pointing outwards, the opposite of our perception of gravity, but the gravity of the dense molten core make it so that the crust doesn't just crumble away.
This is a very improbable scenario. If the force of gravity pulls people and objects away from the molten core, then it is also pulling the crust away from the molten core at the "surface" level. If the crust is very thick, the gravity on the outward facing crust will be stronger than the gravity at the surface. Further, unless the molten core is extremely large then you are doubling its radius very frequently and it is getting weaker and weaker.
If gravity from the stars is pulling people away from the molten core, then the crust should crumble and fall away.
TECHNOLOGY SOLUTION: The crust is contained inside a stronger substance such as a metallic shell which allows it to maintain its structure. Your planet (or at least the inside) would need to be something other than a sphere (cylinder seems likely) or all the people would mercilessly fall to the equator from rotation/gravity. But since the planet was constructed, picking the shape of the interior is easy -- the outside could still be spherical which would be easier to balance with gravity
Heat: Your core is molten rock radiating heat towards the crust, your external surface is heated by constant daylight enough that you say the heat on the surface is enough to prevent light. If there is never darkness then your crust will continually get hotter. If you have heat inside and heat outside, where is all the heat going to keep your people from dying?
TECHNOLOGY SOLUTION: If the external surface is engineered instead of natural (see gravity issue). Then surface could be reflective, sending most of the external light away from the planet. OR it could be an advanced solar array
The Stars themselves: I don't think we know of any natural object with a perfectly circular orbit. For two stars to stay in a perfectly circular orbit such that a planet at the center of the orbit was always in a stable position would be extremely difficult. The masses of both stars would have to be just right, the placement and timing would have to be just right and there would have to be nothing of any size operating in their vicinity to disturb the delicate balance.
TECHNOLOGY SOLUTION: This is a hard one, to move stars into an orbit defies even the craziest ideas we have right now. In the pseudo-science realm, I suppose if you had a race that could manipulate gravity to the point they could create black hole levels of gravitational force then they could manipulate the paths of stars and put them where they want them using basic orbital mechanics. More likely I suppose is that they were so determined to find such a place that they put considerable resources into searching, finding and traveling to the only star system within their galaxy that happened to be right.
Side note: If a builder race could control gravity, the planet could be spherical on the inside and natural outside by making gravity pull towards the center of the crust -- this locally over powers the stars, accounts for all the gravitational issues with the crust/people without an external protective shell, and if finely tuned could be suspending the core in the center of the planet (instead of the core holding the planet together in a delicate balance).
This side note actually has me excited as possibilities go though. Let's say some race of builders long ago (maybe your people and they just lost technology over the eons, maybe a race seeding life, whatever). This race could manipulate gravity. They find two stars in a near circular orbit, manipulate gravity a bit to get them perfect, then construct the planet at the center assuming no one would ever find it.
They probably put solar arrays on the surface to power their gravity technology they build into the crust, then put the core in the center to provide heat and natural light (not enough comes from the stars because it's being harnessed for gravity). The core needs a source of energy itself, but this could perhaps be gravity powered as well, IE using tidal forces or such.
When everything was ready, they put their colonists or whatever in the planet, thinking inside a planet in an impossible location in an inhospitable area of a solar system would never be found and would be able to live safely. An Ark of sorts.