It usually appear redness if a blood vessel to burst in eye. Commonly, subconjunctival hemorrhage is blamed to the burst blood vessel in the eyes. But there is really rare to happen. And subconjunctival hemorrhage is often result of increasing pressure in the body. But it can be serious to suffer it. Personally, you shall take an eye check and consult a doctor for a good treatment to your eyes.
In terms of your description, it is very likely that you get subconjunctival hemorrhage. One of my friends also got this once and his eyes looked very terrible. He was in a panic at that time and then I had to take him to the doctor. The doctor comforted him a lot and repeated telling him that nothing bad will happen. Actually, this is a very common eye disease and there will be several symptoms that come along with when you suffer from this. And sometimes, through these symptoms, you can also be tested if you have high blood pressure or diabetes. Anyway, you have to be very careful, okay?