Is it really possible for a spaceship to stand still and turn off all engines and just be idle in one exact spot in space like it happens in particular famous tv series etc? Wouldn’t the ship just continue floating away/moving on its own until you turn the engines back on?
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No, that is not how physics works.
First of all, everything in space is moving all the time so a complete stop would only be relevant if it is in relation to some other body.
Second, it takes a long time and a lot of power to accelerate and it takes just as much time and just as much power to accelerate in the opposite vector.
The question is, full stop relative to *what*?
There is no zero velocity point in the universe that you can say is not moving at all and the more you look for one the more you learn about how this doesn’t exist. So the “full stop” concept in scifi is closer to “fully matching speed with something else”. And in space it’s totally possible to get really close to zero relative velocity; this is basically what “rendezvous” means in a space context.
If to ships were close together and had their velocities in sync, they would appear to from either that the other was stopped. Movement can only be judged in reference to something. An astronaut that lets go of a ship will not have their velocity in sync with the ship because the act of letting go will cause it be different. Unless the astronaut has a suit with propulsion they will then not be able to catch back up with the ship.
Coming to a stop would be in relative terms to the location of a star or other spacecraft etc. For science fiction tv series and films they normally have a warp drive or similar for faster than light travel, shutting off those warp engines mean that the sub-light drift speed will take them year to travel between stars so for most purposes they have stopped.
With the proviso that “one exact spot” is always going to be relative to something else, sure.
This is, mostly, how things dock with the space station. A capsule gets launched, flies up near the space station, and then uses its thrusters to come to a “full stop” right next to it. Then one of the space station’s arms grabs the capsule and pulls it in so it can dock.
It doesn’t float away / move on its own (or at least not very much) because they’re *REALLY REALLY CAREFUL* about how they use the thrusters so they get it juuuuuust right. Lots of stuff in space travel involves *LOTS* of people *ALWAYS ALWAYS ALWAYS* being *REALLY REALLY CAREFUL*.
An important portion of your question is about *storytelling* in a visual medium. https://tvtropes.org/pmwiki/pmwiki.php/Main/SpaceIsAnOcean and the articles Space Sailing, Space is Air, and Space Friction linked there.
Most of the other comments address the physics of inertia in spaceflight. A lot of famous film and TV science fiction is also classified under [space opera](https://en.wikipedia.org/wiki/Space_opera). There are shows that try to be more realistic with the physics of space. From a storytelling perspective, viewers and filmmakers were both more familiar with the tropes of sea and air travel. Hence why engines are continuously running outside of atmosphere.
Even in water, **station keeping** is needed to maintain position relative to something. [Orbital stationkeeping](https://en.wikipedia.org/wiki/Orbital_station-keeping) refers to the maneuvers needed to maintain a position in orbit or with another spacecraft.
Further reading:
* https://en.wikipedia.org/wiki/Reaction_control_system
* https://en.wikipedia.org/wiki/Space_rendezvous
No, you’re in orbit, so you’re constantly accelerating towards the nearest gravitational body.
In order to maintain orbit, you need to be moving very quickly to the side. So by the time you’ve fallen, you’ve moved far enough sideways that the body you’re orbiting is behind you.
Circular motion requires and acceleration such that a=v^(2)/r
Gravitational force is F=GMm/r^2
Newton’s second law is F=ma
ma=GMm/r^2
a=GM/r^2
v^(2)/r = GM/r^2
v^2 = GM/r
v=sqrt(GM/r) for a circular orbit. G is the universal gravitational constant, M is the mass of the body you’re orbiting, and r is the radius of your orbit.
This doesn’t require any active engines, you just need to get up to speed, and there’s no drag to bring you down. This is how the ISS and satellites stay in orbit, they’re moving very fast.
You can “park” two spacecraft next to each other by making then move at the exact same speed in the exact same direction. They won’t appear to move relative to each other, but they are still moving very fast around the planet. Over time, they will drift towards or away from each other unless you have absolutely perfect precision, but it’s hardly noticeable until several orbits have been completed.
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