See title. This is a shower thought that popped into me while slowly waking up. I’m thinking, what if due to e.g. gravitational shenanigans, the pull on a planet is such that a planet stands relatively motionless in a fixed position towards its star?
Is that possible or am I forgetting some astronomy basics?


You mean a tide locked planet, as in, one side of the planet would always face the star? The Moon does this with respect to the Earth, so yes, it is possible.
If you mean no spinning around the star, then no. The orbital equilibrium is given by the attraction force between the planet and the star being countered by the centrifugal force of the planet spinning around the sun. If the planet were to slow down, it would move closer to the star. If it would stop, it would eventually fall into the star.
What if something was just plopped gravitationally equidistant from two stars in a binary system?
If it was perfectly balanced, sure. Any slight imbalance would cause rapid destabilization.
Also the pure gravitational forces might just rip that thing apart.