Can I get genital herpes without having sexual intercourse?

Yes.

Genital herpes spreads by contact of any type, which exposes you to infected fluid or an active sore or skin that sheds herpes virus. Herpes virus can enter our body through even microscopic cuts in the skin. The virus also enters our body through the mucous membrane. Therefore-

If your mucous membrane comes in contact with infected fluid/infected sore/skin you will get genital herpes. For example if some body having cold sores performs oral sex on you, you will get genital herpes. Similarly when the mucous membrane of your genitals , mouth or anus comes in contact with virus the virus enters our body.

If virus gets in the body through cuts in the skin we will contract herpes.Let us say you touch infected person's genitals with your hand. The virus may enter your body through microscopic abrasions/cuts in your hands.