Why is it that in a quiet room I hear my ears ringing?

No masking. When there is competing environmental sounds, the ringing in your ears can be "masked" or drowned out. This is why some physicians recommend the use of noise machines or air fans for those individuals who had difficulty sleeping due to the ringing when in a quiet setting.
Tinnitus. Ringing in the ears or tinnitus is always present. It is an electrical or flow noise that occurs in the head. When we have eustachian tube problems or hearing loss, these sounds are often unmasked and become apparent. If a room is quiet enough, most people will experience tinnitus. You should also consider having a hearing test to check for unrecognized hearing loss.
Tinnitus. Tinnitus is most often heard when there is no background noise. Medications such as Aspirin or Lasix (furosemide) cause it. Keeping some background noise can prevent it from occurring but you should see an ENT to evaluate if it persists.