Realistic Gravity Manipulation
So, there is a certain level of handwavium element to the execution of this idea, but the physics are kind of solid. Gravitoelectromagnetism. Doesn't have anything to do with electricity or magnets at all, but scientists are kinda jerks like that. Anyways, it was discovered a few years ago by Gravity Probe B that large rotational masses actually cause an effect called frame dragging, whereby space and time are kind of drug along and slightly twisted with the large rotational mass. It was discovered (at least on paper) that two of these rotational masses rotating in different directions could possibly cause some repulsive effects somewhat similar in nature to the effects of electrical charges or magnetic fields repelling each other, except with space, time, and gravity instead.
So basically, in certain theoretical scenarios you can get gravitic fields to repel each other kinda like how magnets or electrical fields do. Also time and space has something to do with it. Its weird. I suck at math too. And physics.
Thing is, we're talking about two planetary masses, and the rotational rate to achieve cancellation has to be nearly relativistic right? So it's not exactly like we can just take two planets and spin them at a high percentage of the speed of light in opposite directions right? Well... kinda.
Take some extremely dense matter from a white dwarf and contain it in some sort of ring. Now you have something as massive as a planet, but just really really dense. Take two of these rings and accelerate the white dwarf matter inside of each ring in opposite directions to a percentage of the speed of light like a really big, really energy hungry particle accelerator.
Viola, opposing frame dragging effects begin. Now here is where things get really cool (Assuming the math is right). Anything sent through the center of these rings will be accelerated without experiencing any G forces. Its basically being pulled and pushed along at the same rate by opposing torsional fields of space and time and thus is in effect, no longer experiencing the effects of gravity. As long as the object sat in the sweet spot between these two cancelling fields it would in essence, just kind of sit there suspended. So you got one of these rings of hyper dense matter being accelerated at high speed below the city, and one somehow above it (in a geostationary orbit maybe?) and the city sits in the sweet spot between the two where gravity is no longer effecting it.
The math is wrong: Somebody might have miscalculated some variables, or may have misinterpreted the findings and the entire theory turns out to be bunk. Frame dragging effects and their exact effects have yet to be accurately modeled and studied, it is believed that doing so would be incredibly difficult and until this is done all of this is still not considered to be confirmed science. The theory is relatively new and not very well studied yet.
Even if Not, Still Waving Hands Pretty Hard: Collecting mass that dense, containing it, then accelerating to relativistic speeds, then keeping it that way indefinitely requires a lot of energy. I illustrate the amount required vs what they are using it for in our terms, its like building a nuclear reactor just to run a single coffee maker. At minimum our race would have to have a Kardeshev 2 level ability to access and manipulate energy. (IE can harness 100% of the energy output of a star or its equivalent.) This level of technology is so unbelievably out of our reach that talking about specifically how they do it with our current level of development is basically waving our hands and grunting "magic." This phenomenon will most certainly never be found in nature or achieved by a pre-technological society. If you want to go that route you are just going to have to utilize the old "its just magic" explanation.
People Dwelling in the City Don't Have Gravity Either Pretty Self explanatory. The object in between these opposing fields do not experience gravity. Period. Nothing in the city is being effected by any gravitational fields outside of the area of nullification. At this point, you might as well just build an orbital city. I mean, maybe your Kardeshev 2 level society just really likes showing off and builds something impractical just because they can, Its been done before. Its just not a very practical way to do things. Maybe if we are really handwaving here the same fantastically advanced people somehow managed to make the field only effect the city, or found a way to counteract it within the confines of the city? In which case you are basically just saying "meh, its science." Then you just end up utilizing the same trope as the magic thing earlier.
Anti-gravity and gravity manipulation may very well be possible, at-least on a theoretical level, but not naturally. The idea that something naturally occurring or predating technological sources could achieve this with magnetic fields or something is bunk. The precisely balanced system of high energy effects required to make something like this happen simply doesn't occur naturally. So I have a proposal:
The great old ones trope. A Kardeshev 2 level society built all these floating cities a long, long time ago as a bunch of resorts or fancy ultra-technological space mansions or something. They all died, or ascended, or got bored and left or something a long long time ago too. Now a bunch of comparatively backwards people descended from their estate servant staff who got left behind live on these floating cities and islands with zero clue as to their origins or original purpose. They've just always been there and as such nobody considers that to be terribly odd or worth looking into. (Also the last time somebody got curios and went poking around they accidentally turned the gravity back on and everybody died so nobody does that anymore.)