Yes, you can. It's just a bit expensive and time-consuming.
You'll need two pieces of hardware, though.
First, a HDCP stripper. It usually comes in the guise of a HDMI splitter. Plenty of them online - look on ebay. ViewHD is one such brand. Just make sure you buy from a reputable dealer.
Second, a HDMI capture device like the Elgato HD60.
Total cost should be under $200.
And then, you just play each video you want to keep from the DVR, and capture the HDMI output through your computer.
Yes, it has to be done in real time.