We are in a house that uses well water and a septic tank. The house had a half-bath designed but the half bath was never finished. There is a pipe for a toilet but there is no toilet attached to the pipe. We had the half bath area carpeted. (Note that the half-bath never had a door put on so it looks like a closet area. Before they laid the carpet down they taped up the end of the pipe. Now if we run the fan in the full-bath on the other side of the unfinished half-bath we smell a raw sewage smell in the half-bath area. Also, if we run the fan in the bathroom downstairs we can smell the raw sewage smell upstairs in the unfinished half-bath. I got down on my knees close to the floor and could not smell the raw sewage smell near the pipe. It seems like the smell is higher up in the room.

I'm not sure if it's related but we have a drain in the basement that we fill with water during the winter.

I am reluctant to rip up the carpet in the half-bath area unless we really need to. Any ideas what the smell could be?

You have to properly cap off the toilet drain. Tape will not do. Also, there is a chance that there is a sink drain in the wall and shower drain as well. Cap the off with glue caps or "dollar" plugs. The way you have it now you are creating health hazard for house occupants.