Well, for your situation, I have to say that you should not be worried too much. And just calm down. And you should know that water and oil do not mix. And if you use water, it will drive the oil in more deeply and can make the burning sensation worse. And according to some researches, you can put vegetable oil at the edge of a napkin and then touch it to the corner of the eye. And also, you can just pour some vegetable oil in the palm of one hand and use the index finger of the other hand to get the oil into the eye. In that way, it will just make your eyes feel better. Finally, wash your eyes carefully. It will be fine.
It is so bad for you to get eucalyptus oil into your eyes by accident. You need to wash it out with clean water as soon as possible. Then you could use the clean cloth to dry the eyes. You could also use the warm compress to make your eyes get comfortable.
Eucalyptus oil is a kind of fluid of colorness or faint yellow, which is usually used to make up cough reliever, mouth wash or essences such as those of toothpaste or candies. You should wash your eyes with clean and flowing water to make the eucalyptus oil out of your eyes. After that, you can use some antibiotic eye drops into your eyes. Pay attention to have good and adequate rest and sleep. After two days of using the eye drops, then you can turn to use some erythrocin eye ointment for your eyes. Then after another two days, if there are no uncomfortable feeling in your eyes, you may set your minds at rest. Just remember that do not put something like this into your eyes in the future. However, if you still feel bad in your eyes after having taken the above measures, you had better go to the doctor's for an eye examination in case that the conditions turn worse.