Yes. Been doing it for years, with no monthly fees, and I love it. I wouldn’t recommend it unless you are proficient at PC troubleshooting, have some experience in poking around inside the case, aren’t afraid to install driver updates, and willing to make changes to the registry manually.
There are lots of options, but perhaps the most common HTPC (home theater PC) setup would include a newer computer (like yours), Windows 7 (which includes Media Center), and a TV tuner card or two (preferably with both ATSC and QAM tuning).
ATSC will allow you to pick up over the air HD broadcasts, and these can be recorded to your harddrive. QAM tuning permits you to pick up unencrypted HD digital channels off of your cable (which will be generally limited to those channels which are broadcast over the air in your area). My local Comcast system just converted more channels to digital (everyone else has to have a small box to tune anything beyond the first 25 channels or so). The upper primary cable channels are also available on the QAM tuner, but in SD only. In this setup, you will not be able to record other channels in HD (like ESPN HD, or HBO HD). You can also get the computer to control a cable box, which will permit you to record premium channels like HBO, just not in HD. The benefit of two tuners is that you can watch one show while recording another.
Another option is to get a Cablecard PC. You can google that to learn more.
hack7mc.com and thegreenbutton.com are excellent resources for further research into a Windows based HTPC setup.