It is possible that you do a same sex marriage in Stardew valley. The creators of game has left the option.
There are twelve eligible people (6 Bachelors & 6 Bachelorettes). You can marry one of them at a time. You can divorce them. You can remarry them after making them forget you were married.
- Your spouse will ask for adoption. If you agree, a child will arrive in middle of 14th night.
- You can change children into doves permanently using the Dark Shrine of Selfishness at the Witch’s Hut, in exchange for a prismatic shard.