Why not? Every DVR I know of has OTA capability.
Not much point of saving to DVD except maybe if you want to give a copy to someone. The whole niceness of a DVR is you get to move away from the limited and tedious VCR concept of having to have stacks of media that you manually change.
No cable necessary. The various do it yourself versions and windows media center all can use a tuner such as a Happauge card which will receive OTA TV allowing you to even record two programs at the same time if you use a dual version. I know tivo can record OTA too.