You could get a computer that runs Ubuntu and put MythTV on it.
Ubuntu Home Page | Ubuntu
MythTV
Or you could run full out Mythbuntu.
| Mythbuntu
As long as you have a decent processor, a decent amount of memory and a decent TV tuner card, you can feed the signal directly into the card if you can get it into a cable type connection from the antenna.
Ubuntu is a version of linux that, in my experience, seems to do a very good job at being a stable operating system. From what I have seen and heard MythTV does a great job at being the DVR you want, including downloading scheduling information. I haven't ever seen Mythbuntu running in person, but it sounds like it just makes the entire computer into a DVR and almost nothing else, only a good option if you have a spare computer that you won't use for anything else.