With the rails and posts in good shape, it's a realtively easy job to remove and replace a section, as long as the new section has the same link dimensions.
If you study the construction of the link mesh, you will see it is constructed of interconnected, flattened spirals that run top to bottom. After releasing the wire ties holding the chain link mesh to the rails and posts, at one end of the damaged section, release one spiral at the top and bottom from the adjoining ones (where they are twisted together) and 'unscrew' it. (You'll need to pull some slack into the section where you are performing this.)
Repeat at the other end of the damaged section, replace with new chain link mesh and reverse the above procedure for installation.
There may be DIY books at your library that explain this better than I, and with pictures, too.
BTW, I have a make-do method of pulling slack into a fence with S-hooks and rope. Your local rental center may have a hook, rope and pulley devise designed for the purpose.