Do I need to have a unit to play Star Crossed?

Yes, you must control a friendly unit on the board to be able to play Star-Crossed.

According to RiftJudge FAQ #10187 and #9452, Star-Crossed creates a mandatory targeting relationship with both a friendly unit and an enemy unit. Because the spell lacks "up to" language, both targets must be present and valid at the time the spell is announced and placed on the chain. If you do not control a friendly unit on the board to serve as one of those targets, the targeting requirement cannot be met, and therefore the spell cannot be played.