String theory has yet to come up with any easily testable predictions despite decades of work. Why is this? Why is it so difficult to come up with testable predictions for string theory?
Why is it still so widely accepted despite the lack of experimental evidence?
One problem is that the Standard Model is too good in the sense that the predictions it makes are expected to work to almost perfect precision for many magnitudes of energy above anything we can produce in a lab environment. As such, no matter what theory of quantum gravity you have, the predicted changes in quantities at somewhat attainable energies have to be minuscule.
On the other hand, string theory is expected to have enough freedom to, depending on the 'exact choice of string theory' (this is known as the swampland), cover considerable regimes of physical quantities such that there probably is not only one string theory consistent with our known physical quantities (up to known precision), but rather a vast amount. (Though actually finding such a theory which is consistent has eluded us so far, if I'm not mistaken.)
So the answer to your first question basically is that it is very hard to find a string theory that is consistent mathematically, consistent with our known physical values and that differs enough for it to be testable at magnitudes of energy that might become attainable in the near future.
So why string theory then? Depending on who you ask, the answer to this may evoke strong emotion, but my favourite answer is that all other proposed quantum gravity models have (some) of the same issues as string theory and also do not predict anything verifiable.
In addition, string theory is very good at naturally producing all known forces and explaining them which some other quantum gravitational models still struggle with a lot more. So the question basically becomes which model you believe can first overcome these hurdles and to me at least string theory is at least as good a candidate as any other and being well studied might even be closest to fixing some of the mentioned issues.
To put it more simply:
The characteristic energy scales at which string theory is dominant are so incredibly high as to require particle accelerators that would need to be ~light-years in length to explore experimentally. This means that direct experiments on strings will never be possible with machines that humans could ever build.
String theorists are hence limited to searching for any string effects that are left over in the low-energy limit of the string model and thereby within the capability of our test machinery to probe. The problem is that there aren't any string effects in the low-energy limit because the low-energy limit is designed to be the Standard Model, which doesn't contain strings.
Here is my take on your question. I might come back to it when I have more time hence the square brackets at the end. I address the answer at the level of an enthusiast who is not technically minded, but have tried not to hide difficulties.
Your question implicitly contains two rather different questions:
Why is string phenomenology (the study of the relation between string theory and the real world) in such a primitive state at the moment that it mainly is interested in the questions: can the standard model be embedded in a string theory; can string theory give rise to a universe with (at least what appears to be) a positive cosmological constant; in other words is the idea of string phenomenology even viable, sweeping aside practical experiments?
Why does virtually every theoretical physics faculty contain people who routinely employ string theoretic methods?
I get the sense that your interest is probably in (1), but let me begin with a few words about the difference between these two points. There are two broad ways in which one can do physics. You can either perform an experiment and subsequently develop a corresponding theory to explain the results. Or you can try to understand the structure of known physical theories, exploring how fixed this structure is and what happens when you change some aspects. (In reality, physics often develops pretty randomly, rarely fitting the idea of 'experiment then explain', which is why there are so many books on the history and philosophy of physics).
In high energy physics, the current state-of-the-art theory is the standard model (SM). It is an example of a quantum field theory (QFT). As exposited in the previous paragraph, there are two broad approaches to research related to the SM: either build a large accelerator to investigate whether new physics exists beyond the standard model, or study quantum field theories more abstractly to try to understand them better - perhaps this will bring more understanding to the SM as a particular case, providing perhaps a new idea for an experiment, or allowing new predictions to be made. Physicists often say that they don't even really know what a QFT is! A prototypical example of this second approach is the study of quantum chromodynamics (QCD), which is badly understood in a certain 'strong coupling' regime. It turns out that all QFTs have the same kind of difficulty, so it seems sensible to study simpler QFTs with the hope that you might have more success which will be transferrable to QCD.
In practice, for most physicists, string theory is used as an umbrella term encapsulating an absolutely enormous framework of interlinked ideas and methods which have links to string theory. Many physicists don't 'accept' or 'reject' string theory - they routinely borrow such ideas for their own purposes. It is worth noting that many of the major theoretical advances in physics in the last couple of decades have arisen in such a way, benefitting enormously from string theory, ranging from supergravity and supersymmetry, supersymmetry breaking, non-commutative geometry, geometric engineering, non-Lagrangian theories, extra dimensions, holography, generalized symmetries, various aspects of cosmology, UV/IR mixing, the swampland program, to many more depending on your preference. Because string theory is a theory of quantum gravity, and is widely believed to be the only such theory which really is viable, calculable and which provides fresh ideas for QFT, one can also use it to learn about or at least suggest ideas for general quantum gravity theories. All of this constitutes one of the largest research programs in physics ever, providing continuing successes. In many cases, one is only scratching the surface, and this entire breeding ground is likely to continue giving a lot more, at least this is what most physicists who work in the area feel. This is some rather wordy answer to 2. above.
As for 1. I agree string phenomenology is in a very primitive state right now, but there are many people working hard to try to improve it! The basic problem it has is likely to be shared by any theory of quantum gravity: the scale where quantum gravity must take over is $\Lambda_{pl} \sim 1/M_{pl}$ which is of the order $10^{19}$ GeV. To get an idea of the enormity of this number, note that the standard model is tested up to energies of around $\Lambda_{SM} \sim 10^{4}$ GeV. Thus, unless by some miracle the quantum gravity theory steps in a lot earlier, most QG predictions at CERN energies will differ from the SM by factors of $\Lambda_{SM}/\Lambda_{pl} \sim 10^{-15}$, which is probably not testable, or at least no one has an idea of such an experiment at the moment. Any string theory is a physical theory and so does of course make predictions, but they will typically suffer from this problem. With the recent advances in cosmology experiments, there may be some hope of probing quantum gravity in the not too far future, but time will tell.
The other problem string phenomenology has is that string theory appears to have an enormous landscape of vacua. It is worth pointing out that this space of vacua, despite all claims about its enormity, has measure zero compared with the space of all QFTs. The current swampland program is an excellent idea to try and understand precisely which QFTs cannot come from a string theory (or in the stronger form of the program, which QFTs cannot come from any theory of quantum gravity). Actually, it is even worse: there are other string theories like sub and supercritical string theory which are so difficult to study that they are rarely discussed at all, but are logical. In some sense, you should think of string theory has a framework like QFT - it provides many possible candidate theories/vacua, and your task is to find one which matches on to reality. The major difficulty is sorting through this landscape - it is not even clear whether the SM is contained in it. A good deal of string phenomenology is related to this question, with modern machine learning methods becoming increasingly important. Once one has found a string theory vacuum which could potentially contain the SM and also a physically sensible cosmology, the task would then be to try to be more creative in coming up with ways to test it.
[to put in: an example scattering amplitude to show the scales where QG is important? Polchinski reference? Add SM Lagrangian with higher order operators]
Why is is so hard to come up with testable predictions for string theory?
There are two main reasons. The other answers have mentioned that there is a very large number of possible worlds to test in string theory (differing by shape of the extra dimensions, arrangement of "branes" and "fluxes", etc., all of which have consequences for how the strings behave).
But the other big problem is that, even if you choose just one possibility to work with, it is very hard to calculate anything that might actually be measured, like the masses and the couplings of the particles. (I linked to a detailed example in a recent answer.)
Why is it still so widely accepted despite the lack of experimental evidence?
It very naturally gives rise to the kinds of fields that we see in nature (fermions, gauge fields, gravity), and to further phenomena that have been widely anticipated by particle physicists (grand unification, supersymmetry). It provides viable answers to theoretical problems of quantum gravity (how does physics work at the Planck scale, where does the entropy of a black hole come from). It is also a very fertile mathematical laboratory for problems of field theory (confinement, duality, scattering formulas).
So string theory has the "feel" of a theory of everything, but even now, it's more at a stage of model building, rather than model testing. It's a victory if you can find a string model which captures one more qualitative feature of the real world, like a heavy top quark or a small dark energy.
In that regard, the fact that the Higgs boson showed up without any superpartners is evidence that a lot of string model building has been on the wrong track, and that much more attention should be paid to non-supersymmetric strings. I will embarrass myself by making a meme out of it...
The problem with string theory is that it cannot make proper predictions. Originally, there was much excitement when string theory was developed. But then it turned out that there is an astronomical number of different versions based on how the extra dimensions can be compactified. Perhaps one (or a few) of these versions would be consistent with the standard model, but there is no way known to find the right one. As a result, there is no known way to make testable predictions with string theory.
A theory that cannot make predictions is said to be not falsifiable, because one can never know whether it is a valid description of nature. In that sense it is not considered to be a scientific theory.
