What is the obstacle to writing down a String Theory that describes our universe?

Even though string theorists have constructed huge classes of exciting semi-realistic compactifications (see this for a modern example), the problem is our lack of understanding of how to systematically achieve string theory compactifications that share properties with the universe we observe.

The paper Life at the Interface of Particle Physics and String Theory is an excellent divulgative discussion on the problems and advantages of many specific string constructions.