I'm not so bearish on this idea as some of the other commentators.
In Loop Quantum Gravity (the smaller of one of a couple main efforts to develop a theory of quantum gravity), one of the core concepts is that fundamentally space-time is made of points connected to other points and that notions like "locality" and the number of space dimensions are only "emergent" properties of space-time at a macroscopic level that arise from points that have links to each other in common being perceived as being close to each other.
So, at a macroscopic level, space-time seems to have three spatial dimensions, to have well defined locality, to be smooth and to be continuous, but really, space-time is a network of discrete discontinuous points with ill defined dimensionality, in which it is entirely possible for a point to have a direct connection to another point light years away, even though most of its connections, on average, will be very near by.
In LQG, the true formulation of the speed of light is not in terms of distance per time in a continuous space, but in space-time points traveled per time whose average apparent distance from each other is well defined, but which actually have variable apparent distances from each other according to a well defined probability distribution that is reflected in the fact that accurate quantum mechanical calculations for the travel of a photon, for example, require not just consideration of all possible paths from point A to point B, but also a range of speeds above and below the speed of light (which is just a most probable average speed) that in an LQG version of quantum gravity may reflect scenarios in which the same number of space-time points are traversed in a given time frame, but the distances between them only produce the speed of light, on average, rather than exactly replicating it.
So, in a universe with non-local space-time connections, there is no intrinsic difference between going from one side of an atom to another through an apparently local connection of space-time points, and going from Earth to a distant galaxy. Both could involve the same number of space-time hops.
These non-local connections and the very notion of gravity itself, in some versions of LQG, are deeply related to the quantum mechanical phenomena of "entanglement" which allows particles whose wave functions collapse at greatly separated locations in time and space but once shared adjacent points in space-time to continue to act in a corrolated fashion.
Now, the fictional side of this when you get to a macroscopic sized gate based upon current technology is that in LQG and with respect to the quantum mechanical phenomena that seem to be a good fit for this kind of space-time, non-locality is a phenomena that generally happens on a one point in space-time just big enough for a single photon or quark or electron to occupy at a time basis, not on a macroscopic basis where everything averages out and creates the spatially three dimensional world in which concepts like locality and a smooth space-time and perfectly invariant speed of light seem to describe reality.
Essentially, without to much handwavium, the advanced technology that creates the gates would have to figure out how to get all of the space-time points and fundamental particles that make up the gate into a single unified quantum state with a parallel entangled set just like it, and then once the entanglement is established, to ove the gates far apart from each other. And, you would have to devise some way for this intensely coordinated collection of gillions of fundamental particles in two main clumps at the side of each gate, to do so in such a manner that someone passing through it is neither assimilated into the gate's coordinated wave functions itself, nor disrupts the epic feat of quantum mechnical coordination.
Realistically, even in the most far out versions of LQG (assuming that this proves to be correct rather than the alternatives), at best, it would merely be possible to exchange very low bandwidth information across non-local space-time connections that scientists discovery that send photons to known non-local destinations through non-locally connected space-time points, rather than any macroscopic gate.
But, probably the closest to theoretically possible science way to imagine such a technology would be to think about the interior of the gate (the frame of the gate would contain the "active ingredient interior" of it, so you could move it around) as some sort of near perfect Bose-Einstein condensate entangled with another near perfect Bose-Einstein condensate (each of which can only exist sustainably in very extreme conditions) in another gate.
Perhaps, somehow people and thinks can cross unscathed because passing through happens in such a tiny time frame, perhaps on the order of 10^2 units of Planck time (which are 5. 562