Possible. Plant leaves contain chemicals that can lead to an allergic reaction. Additionally, food crops may be sprayed with materials which can also lead to rashes. Contact allergies generally result from repeated exposure to a substance. In the future, wear gloves or some other protective barrier to limit exposure. Additionally, wash well after contact. See your doctor if persists or worsens.
Sure. People have contact allergies to just about anything, including water and sunlight. Don't forget, though, that you can also be allergic to things that have been sprayed on strawberry leaves, such as pesticides and fertilizers. It might not be the leaves themselves.