OK, just go to Recorded TV library and see if there are any channels that are being recorded without you knowing it.
I also use Windows Media Center for watching live TV. Its normal for Windows Media Center to eat up space on the hard disk while watching TV, have you noticed that its possible to rewind the live TV you are watching on Windows Media Center as long as you don't change channels? This means that what you are watching on the live TV is temporarily being held up by the drive but when you change channels or close WMC, this is reset and the space is freed again. My hard drive is a bit small is capacity but I have not encountered any problems of full space warnings like that. My advise to you would be switch channels every few minutes because this frees up some space.
Even if you are not recording any channels, have you set the recorded TV storage to another drive besides drive C:? In my observation, WMC uses the recorded TV storage drive as the same drive where it temporarily holds live TV data. So if your recorded TV location is drive D: but then drive D: has little space left, then you will have a problem. You will have to change the recorded TV storage to another drive that has more available space so that drive will also contain live TV data as you watch TV.
Try to confirm this for yourself. If you are still having problems, you could try resetting Windows Media Center setup. Also check up your hardrive from My Computer just to see if it really is lacking space.