Can you play Star-Crossed if you don't control a unit?

No, you cannot play Star-Crossed if you do not control a unit on the board.

According to RiftJudge FAQ #9865, #10187, and #10769, Star-Crossed creates a targeting relationship with both a friendly unit and an enemy unit. Because the spell lacks "up to" language, both are mandatory targets. You must control a friendly unit on the board to serve as a valid target for the spell at the time it is announced and placed on the chain. If you do not have a friendly unit on the board, the targeting requirement cannot be met, and the spell cannot be played.