Put a human being up in space, away from the gravitational bonds of the surface of the Earth, and they'll experience weightlessness. Although all the masses in the Universe are still pulling on them gravitationally, they pull on whatever spaceship you're in equally, too, and so you float. In TV shows and movies like Star Trek, Star Wars, Battlestar Galactica and many others, though, you always see the ship's crewmembers stably on the "floor" of the starship, regardless of any other conditions. This would require some type of artificial gravity to be physically possible, but that's a tall order for the laws of science as we presently know them.
For gravitation, the big lesson from Einstein is the equivalence principle: that a uniformly accelerating reference frame is indistinguishable from a gravitational field. If you were in a rocket ship, and unable to view the Universe outside of your surroundings, you'd have no way of knowing which one was going on: do you feel the downward force due to gravity, or do you feel the downward force because your rocket is accelerating in a particular direction? That was the very idea which led to General Relativity, and more than 100 years later, it's the most correct description of gravitation and acceleration that we know of.
There's another trick we can use if we like: we can make a spaceship rotate. Instead of a linear acceleration (like a thrust on a rocket), you can have a centripetal acceleration at work, where a person on board will feel the outer hull of a spaceship pushing them towards the center. This was used famously in 2001: A Space Odyssey, and would, if your spacecraft were large enough, be indistinguishable from a gravitational force.
But that's absolutely it. Those three types of acceleration — gravitational, linear, and rotational — are the only ones we have that will have the effects of gravity. Which is a big, big problem aboard a spaceship.
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