Regarding the original question about Kaluza-Klein and the standard model:
Up until World War 2, the only known fundamental forces were gravitation and electromagnetism. If we focus just on the force fields, the simplest version of Kaluza-Klein unification - 5d Einstein gravity, with the fifth dimension forming a small circle - gives you 4d gravity, an electromagnetic field (from the metric components relating the first four dimensions to the fifth), and a new scalar field corresponding to the size of the fifth dimension.
Postwar physics eventually yielded knowledge of two new forces, the strong nuclear force and the weak nuclear force, described by generalizations of Maxwell's equations, the Yang-Mills equations. And it turns out that if you have seven extra dimensions in the right shape (a 7-dimensional manifold whose "isometry group" is the same as the "gauge groups" of the standard model, SU(3) x SU(2) x U(1)), you can get *all* the non-gravitational force fields from components of the 11-dimensional metric (in effect, from the quiverings of the 7-dimensional manifold). It's an amazing thing because 11 is the number of dimensions in one of the natural candidates for theory of everything, maximal supergravity (which today we would understand to be an approximation to M-theory, the 11-dimensional version of string theory in which all the strings have become 2-dimensional blobs called M2-branes).
However, the Achilles heel for Kaluza-Klein has always been matter. In its original form, when it wasn't even quantum mechanical, matter particles were imagined to be point particles following the 5d version of the "geodesic equation", the equation in general relativity which describes e.g. how orbits arise by following the shortest path in curved space. The concept was that a particle would be moving through the 4 large dimensions, and around and around the 5th dimension, and charge would be related to that 5th-dimensional motion. Interpreted thus, the part of Maxwell's equations which describe how a charged particle responds to the electromagnetic field are reproduced, but the actual values for charge and mass are related in ways that are completely wrong.
Meanwhile, when it comes to the nuclear forces, the interaction of a particle with the Yang-Mills fields is described by Wong's equations. I don't actually know for sure if geodesic motion in the multiple extra dimensions of these more complicated Kaluza-Klein spaces, reproduces Wong's equations. Probably it does, but at this point we can't ignore quantum mechanics any more, e.g. the interactions of quarks with gluons are very quantum-mechanical.
So instead of looking at the geodesic motion of a point particle, we have to consider fermionic quantum fields, from which matter particles arise as quanta. We also want these matter particles to be fundamentally massless, since that's how the standard model works - the massive particles we observe, consist of massless left-handed "Weyl fermions" coupled to massless right-handed Weyl fermions via the Higgs field, producing massive "Dirac fermions" with both left- and right-handed states. It's a little intricate but that's how we understand what experiment shows us.
This means that in a quantum Kaluza-Klein model, instead of trying to fiddle with the geodesic equation so that the masses and charges come out right, you want the Kaluza-Klein "zero modes" of your 11-dimensional fermionic field (the zero modes are a kind of harmonic of the field in the 7 compact dimensions) to have the correct interactions with the metric (in particle physics terms, they must belong to the correct representations of the gauge groups, to match the matter fields of the standard model).
And *this* is the thing that no modern Kaluza-Klein theorist ever managed to work out. The fundamental problem is that in the standard model, the left-handed and right-handed Weyl fermions interact differently - the left-handed ones interact with the weak force, the right-handed ones do not. But the components of the 11-dimensional fermion field, when compactified, must behave the same, regardless of their 4-dimensional handedness.
So you may ask, what's going on with all the extra dimensions in string theory, then? The answer is that they are not being used in that way. In a true Kaluza-Klein theory, the shape of the extra dimensions has a symmetry (an "isometry") and the non-gravitational force fields are produced by vibrations of the extra dimensions. The Calabi-Yau spaces used in string theory do not have isometries, they don't have that kind of vibration. They do have size parameters which can be excited, producing particles called "moduli" which remain completely undetected. In string theory, the non-gravitational forces are produced in other ways possible only for strings rather than point particles (either by "open strings" whose endpoints are attached to parallel branes, or by special closed strings that carry charges on them).
However, Kaluza-Klein behavior is possible in string theory, if the extra dimensions do contain an isometry after all (e.g. if one of them forms a circle that can expand and contract). And indeed, similar to what
@Orodruin mentioned, there is a class of string theory models in which one of the extra dimensions (called the "dark dimension") behaves that way, and thereby produces dark matter.
Now I should say that Kaluza-Klein theories have never been completely abandoned. The concept is there as a permanent possibility, and there's always a few people still trying to make it work. But when the modern wave of interest in extra dimensions began in the 1970s, the concept was deeply explored and the mainstream of model-building abandoned it, essentially because of the problem of how to get matter that behaves differently according to its handedness ("chiral fermions"). It must have been hard to abandon because there are just some really striking properties of isometric compactifications of 11-dimensional supergravity. The simplest one, compactifications on a 7-sphere, produces "N=8 supergravity" in four dimensions, a theory which comes tantalizingly close to the standard model in certain respects. People really wanted one of those 7-manifolds with a standard model isometry group to do the job, but the problem of the chiral fermions seemed to be insurmountable.
One attitude is to say, well, string theory or M-theory is probably the real theory of everything, and it's a very rich theory, it contains the potential for Kaluza-Klein behavior but it also contains so much more, and it would be a mistake to get sentimentally attached to the idea that the non-gravitational forces (the gauge fields) should come from Kaluza-Klein and nothing else. Modern theoretical physics allows for a very large number of possible constructions; many of them have very appealing or striking features; but only one of them can be right. Maybe that neat coincidence that the standard model needs seven extra Kaluza-Klein dimensions, and M-theory has four plus seven dimensions, is just a coincidence. Or, maybe there's an unrecognized loophole and you can get chiral fermions after all. (Edward Witten, in a highly cited 1981 paper which goes into the problem, concludes by speculating that an additional effect called torsion might make a difference.)
