I Is there any 2D surface whose metric tensor is eta?

Yes. Two-dimensional flat spacetime. Or the worldsheet of a string. What are you looking for in particular?

 

Your question is unclear. If you are asking if you can get that metric on a two-dimensional surface induced by its embedding in a higher-dimensional Euclidean space, then no. The metric tensor induced by an embedding in a Riemannian space is going to be positive definite.

 

You mean ordinary space? But you have a Lorentzian signature. Seems to me like looking for complex Majorana spinors.