It is one of the worst feelings in the world. BUT...it's the worst feeling for the OWNER. For the furry critter, it is almost ALWAYS relief. They don't know they are mortal and when they are euthanized, they don't know it's the end of the road here on earth. I know that doesn't lessen the pain for YOU any more, but often it's the last loving thing you can do for them.
A couple of days ago, I was at the vet and another client had a young dog that looked EXACTLY like a mutt that we had to euthanize 9 months ago at the young age of 12 years because of cancer. The emotions of that dreadful day were overwhelming and I miss that dog more than I would ever imagine, but I know sending him to the "rainbow bridge" was the right thing to do.
Edit: I wanted to add that the "at home service" is really the way to go. Over the years, I have had to put down a number of critters and never thought about doing it at home. With the last one, that dog HATED the vet and was always scared of the experience so I thought I owed it to the dog to make it as painless as possible. Prior to the cancer diagnosis, he had gone to the vet at least 6 times in the last few months because of a CCL surgery. Having the "last act" done at home was more relaxed for the pup and for DW and me. Plus, we didn't have to make the painful march through the vet's waiting area (and the drive home) while crying like a baby.