I am doing this with an old Trek 720.
Mine is a 1984 and in working order so I am in no rush to replace most the parts. I commuted to work on it today and it was fine.
The condition of the old bike you have will be one deciding factor. If you can get away with replacing very little then it is cheaper to do that.
If you have to strip the frame and replace everything then it will cost as much or more as a new bike.