Is there a way to tile only a part of a texture? So for example, I have this eye texture, and I would want to tile only the area marked with red, so that I would't have to have a high res texture wasting my limited hard drive capasity, which has only a small eye in the middle?
This is not possible the way you describe it. What you would usually do is either one of: 


If the rest of the eye is only a single color, you can rescale the UVs for these parts and even overlap them. (In this example you could also cut off most of the white in the texture)

If the rest still needs more than a single color, but less details, and you want to use tiling, then you should use two different textures.