If your fiction needs it, make it so.
Such artificial gravity works perfectly well in Star Trek, Star Wars, Battlestar Galactica, etc. ad nauseum. No-one needs to explain how it works (nor can they beyond techno-babble).
Maybe someday we discover a way to do so in the real world. Not that we have any promising loopholes in physics that we know of that would allow it.
Others have already pointed out the ways to simulate gravity via rotational or linear acceleration, or the (highly impractical) use of degenerate matter plating to create real gravity.
Just write your story with real "Star Trek artificial gravity" and move on. Drop in the techno-babble if you think it makes your story more interesting, enables a plot point, etc.
Some have complained, (I believe unfairly), that I did not answer the question, so here is my alternate technobabble, one that I've never seen used in fiction - but if someone has, be sure to point it out.
It is actually based on known physics - or at least plausible for a story. Einstein's famous equation declares mass energy equivalence, and this works both ways - energy warps space too.
This part is real physics. Much of the mass of some particles is not mass per see, but the energy associated with the energy binding the particles together. E.g., 6 Deuterium atoms has more mass than one C12 atom due the the differences in binding energy, etc. Both are 6 neutrons and 6 protons.
Imagine an incredible superconducting network that resides below the floor or your space ship - one with properties far, far in excess of known superconductor current densities. Pump it full of incredible amounts of current and you now have a electrical "mass field". Extend the network over the entire hull of the spherical ship and you have a way to divert the mass field uniformly into the hull so that it has no net gravity field in the interior of the ship. This allows you a way to modulate the level of energy in the floor plating by switching the energy between the floor plate and the hull. Unequal distributions of the mass field allows creative custom gravity configurations.
If you don't think energy would work this way, enlighten yourself with the kugelblitz - a concentration of energy so great that it creates a black hole. Back in college many years ago, I actually came up with the idea of a kugelblitz, only to later discover that it was first published in 1955 as Geons by John Archibald Wheeler.
What you don't want to consider too closely is just how huge a supply of energy will be required. Almost 90 billion MJ (9E16 J) to create a mass field of 1 kg, and you need at least trillions of kg of mass to get a decent level of gravity for your ship. Which means that starting and starting your ship with trillions of kg of energy is not going to be easy.
For comparison, 9E16 J is about 12 hours worth of electric power for the entire world. Or about 1/2 the energy released by the Tsar Bomba
So, I propose adding a 2nd handwavium - the ability to create and destroy energy (from the perspective of the ship) by using zero-point energy (ZPE) as a source/sink for loading and unloading the superconducting network.
Such combination of technologies also gives you a way to escape the tyranny of the rocket equation. Because you can "create mass" by using the ZPE converter as a mass source you could create regular mass to eject at high speed for acceleration. Of course, if you can do this, you could also simulate gravity by the simple expedient of continuous 1 g acceleration.
You need more handwavium to explain the structural strength of the ship needed to support the use of such mass fields, but such super materials are also known in SF.
The problem with this solution is that it is complicated and boring. If you get the engineering contract to produce the mass field for the latest generation of ships - it would be exciting indeed.