There is no simple polynomial that interpolates that curve. It clearly has a singularity at zero. The last time I checked, polynomials have no singularities.
Next, interp1 gets upset because you have at least some points that have the same x value. If two points have the same x value, but more than one possible y value, how do you expect interp1 to return a result? Which of that y values should it return?
Another possibility is that you might not have replicates, but that your curve has noise in the x variable, so that the curve appears to be non-monotonic in x. Again, it leaves you with a function that is not single-valued in the form y=f(x). So again, interp1 will not function.
The solution is usually to average the y values if you have replicates, replacing the replicates with a single point at that mean value. interp1 will now be happy. If it is just noise that made the x values non-monotone, then you might do a variety of things.
No matter what you do of course, you still won't get anything that is polynomial-like from that curve.
And since you have not attached the data itself, I cannot even suggest a viable model for this. (I can think of a few.) Really though, it is you who needs to pose a model, since only you know where the data comes from and what it means.