Reproducing a Fiber Optic Shape Sensor with Arduino?
I think nobody here can really give you an answer, if this is possible with an Arduino, since we don't know enough about the technology and the nitty gritty things, that make this a difficult task. So making a hobbyist version won't be easy and maybe really expensive.

In the fiber each bragg grating reflects a bit of the light back. The wave length of this reflected light is dependend on the angle between light and grating. So you get a specific wave length back from each grating, that corresponds to the bending at that particular part of the cable. The task is to measure the many wave lengths, that are being reflected in the fiber, and match them to a particular part of the cable. Since they don't say, how they are solving this, I will mention some ideas here:


Spectral Measurement (Wave length):


From the video I guess, that they use white light to shine into the cable. The gratings will only reflect the wave lengths from this spectrum, that satisfy Bragg's law due to bending angle. You have to measure the wave length directly with a spectrometer. Mostly this is another grating, that will seperate the wave lengths spatially, so that you can measure the spectral structure by using a spatial sensor, for example a camera or a line sensor. You would have to correct for different wave length sensitivities of the camera.
Maybe they are only using a laser with 1 wave length. Then the bending would result in a change in intensity reflected from that grating. If you can seperate the signals from the different gratings good enough, you would be able to measure just the change in intensity, which would be easier, since you basically need just a photo transistor or similar for that wave length. (In this case you would need to calibrate the sensor each time as a straight cable and ensure, that the measurement is fast enough for the expected movement).

Matching of signals to each part of the cable: Here I guess, that they use a time-of-flight sensor, because I cannot think of anything else without further information. This is a really difficult problem, because most available sensors are not sensible enough for measuring quarter inches without being cheap in the first place. You will likely spend a lot of money for a sensor like this. And you building an own time-of-flight sensor with the wanted sensitivity may be as difficult as building the whole device in the first place.
Needed Parts: Also you will need the fiber itself, which is - as stated - not a normal glas fiber, which you can by everywhere. It is most likely a custom build and you will pay a lot of money, to get one of these. The light source is dependend on the spectral measurement above. For a white light it is important, that all wavelength should have as equal intensity as possible. You would also have to account for the spectral absorption and the overall absorption in the fiber, which restricts the sensitivity and possible lengths.



  Is there anywhere to buy this sensor directly?


Most likely not. In the videos it seems like a device, that was developed by the NASA for a specific application. The video is a proposal for other companies to learn from the NASAs development, so the device is not consumer ready. But you might be able to get a paper about this device by searching the internet for it or by asking the NASA directly. This paper would contain way more information about the device as the videos.

All in all this would be an extremely difficult task. Since you came here (a simple Q&amp;A site) with this difficult question, I think, that you are not really ready for this big of a task. Most likely even the developers at the NASA needed long and much money for this.




  Are they any alternatives for 3D shape/position sensing besides 3D magnetic sensing?


I would propose to build a string of bending strips, which change their resistance according to the bending angle. You can use multiple strips for getting multiple readings for different parts of the human arm. With each strip you build a voltage divider and do an analog measurement. With a bit of calibration you should be able to measure the curls.