My house had over 1000 square feet of oak flooring under carpeting when I bought it. I am refinishing this. My problem is in the center of this, near the entry way, there is a small section (100 square feet) that did not have hardwood and is surrounded by hardwood on all sides. How or why this happened is a mystery.

I could just put down hardwood in the small section where it is missing, but you'd have a line where all the boards and end start around this area. I was planning on lifting up the old hardwood floor next to this section, so I could redo it seamlessly.

This is very difficult. It is impossible to take up 1 board without taking up the boards next to it, it is difficult to take a board out without damaging it, and it is almost impossible to cut through solid oak with a chisel, let alone do it and leave a clean edge.

Does anyone have any advice for me? Is it practically impossible to patch existing floor? Should I give up?

You should lace it in. Is it not Impossible to do if you have the right equipment. First you want to choose which boards you want to take out to remove all the lined up joints at the end. I usually put a x on boards I want to pull. Then get out a skill saw set the depth to the height of the floor, this is important because you don't want to cut your subfloor. Then you are going to rip both sides of the board right next to the edge all the way down to the next joint. Remember you are going to cut into nails. Once you make the cuts the middle should be fairly easy to take out. Then you can chisel the rest of the board fairly easy. Sharpen your chisel!!!!! If the board is long and you don't want to take out the whole thing use a square and make a new but joint. This you will have to chisel.