There is an extensive and very sympathetic description of a same-sex relationship in the Bible, the story of David and Jonathan. While their bond is described as non-sexual, it is difficult to characterize it as purely one of friendship.
Jonathan was the son of Saul, David's nemesis. Their souls are described as 'knit together'. David and Jonathan 'made a covenant, because he loved him as his own soul.' The word convenant is significant, because in the Tanach this word always implies a formal legal agreement. To mark this convenant, Jonathan literally gives David the clothes off of his back, as well as other gifts such as weapons.
Later in the narrative, Jonathan successfully intercedes with Saul to spare David's life. At their last meeting they are described as kissing one another and weeping together. David's grief at Jonathan's death is profound and moving. In Davids lament for Jonathan he describes their friendship as '(sur)passing the love of women'. This elegy, known as 'the Bow,' is one of the most beloved passages in the Hebrew Bible.
This narrative far outweighs the two trivial aspersions against same-sex love in Leviticus. The bigots who use the Bible to assault gays are apparently blind to it.