Usually,when you get subconjunctival hemorrhage, no treatment is needed, and it will take about 3 weeks for the redness to go away. It is not contagious, you just not rub your eyes and avoid lifting heavy objects. When it is appropriate, you can wear safety goggles. Just do this and your subconjunctival hemorrhage will go away and it will not be contagious.
As far as i know, subconjunctival hemorrhage are also known as hyposphagma. It is not contagious. In fact, subconjunctival hemorrhage is bleeding underneath the conjunctiva. And conjunctiva contains many small, fragile blood vessels that are easily ruptured or broken. It sounds really terrible. If you suffers subconjunctival hemorrhage, you'd better see a doctor. Best wishes.