I have a texture on my model and the problem is that when I made any changes on one side it appears on the other. I believe the reason of that is that while unwrapping my parts of model doesn't fit on the texture. One way I could think of is to increase the dimensions of texture but to make it as big as I wan't would make the size of picture around 20GB. The other is to reduce dimensions of my model parts but that way I would lose quality and it won't look how I want. So I need that the texture would repeat independently
All those islands should be in the 0-1 space. There are some reasons to move things out of that area, as an example, if you want the texture to repeat or if you want to exclude part of a model from a texture bake operation.
For what you are doing there, I think it's best that you do what is shown in the following image. While in Edit mode, move all the UV islands into the light grey area that is displayed with the grid.
The dark grey zone is out of bounds.