What evidence exists for string theory viability?

Some random points in support of ST, with no attempt to be systematic or comprehensive. I will not get into a long discussion, if someone does not find this convincing I'd advice them not to work on the subject. I also don't have the time to elaborate or justify the claims below, just take them at face value, or maybe you can ask more specific follow up questions.

  1. ST incorporated naturally and effortlessly all of the mathematical structure underlying modern particle physics and gravity. It does so many times in surprising and highly non-trival ways, many times barely surviving difficult consistency checks. Certainly anyone with any familiarity with the subject has a strong feeling that you get out much more than you put in.

  2. ST quantizes gravity, and that form of quantization avoids all the difficulties that more traditional approaches encounter. This is also surprising: it was developed originally as theory of the strong interactions, and when people discovered it contains quantized gravity they spent years trying to get rid of it, with no success. As a theory of quantum gravity it passes many consistentcy checks, failure of any of them would invalidate the whole framework, for example in providing a microscopic description of a large class of black holes.

  3. ST extends the calculational tools available to us to investigate interesting physical systems, many times the only such calculational techniques available. Again, it does so in novel and unexpected ways. It therefore provides a natural language to extend quantum field theory, to models which include quantized gravity, and (relatedly) models with strong interactions. Many calculations using that language are simpler and more natural than other methods, it seems therefore to be the right language to discuss a large class of physical systems.

  4. ST contains in principle all the ingredients needed to construct a model for particle physics, though this has proven to be more difficult than originally thought. But, in view of the above, even if it turns out not to provide a model for beyond the standard model physics, it is certainly useful enough for many physicists to decide spending their time stuying it.

Of course other people may have their own reasons to find the subject interesting, I'm only speaking for myself.

The main motivation is that string theory incorporates gravitation and gauge theories in a unique framework, avoiding the problems of General Relativity and Quantum Field Theory.

Besides this basic fact, I would say that a great theoretical success of the theory is black hole physics. For the first time we have a quantitative framework in which to do calculations (scattering, emission) and to see the quantum structure, the microstates, of the black hole. The entropy of these black holes, constructed for instance with a system of intersecting branes or even simply with fundamental strings, nicely matches the expected Bekenstein-Hawking Area Law.

The missing step to achieve seems to be technical. For black holes, we are not able to do the crucial calculations in four dimensions without unphysical assumptions (unbroken supersymmetry , extremality). For the standard model, we are still not able to find the right flux compactification or intersecting brane model to give standard model physics. For general background metric, we are not able to quantize the string. The full non perturbative structure is still work in progress (M-theory...).

A one proposition summary for this sketchy answer: String Theory is considered viable because it's a working theory of quantum gravity and the feeling is that with further work it will became possible to relate theory with our world, that is to get predictions and experimental confirmation.