At the end of August I had given a $850 security deposit for an apartment that I was moving into September 1st. A few days after I had given the deposit, the apartment flooded. The carpets where soaked and the water came in behind the kitchen cabinets. I was told that they had to fix everything and I could move in Oct 1st. In the meantime, something had come up and I am unable to move in to this apartment. I was sprung with this information on Sept 26th, which only left 5 days before I was supposed to move in. No contract or lease was signed and now they won't give me my security deposit back. Can they do this?

No, they cannot do this. Send them a letter by certified mail saying that they have failed to deliver the apartment as promised and that you had no choice but to find another place to live. Then say that they must return the security deposit in full immediately. If they don't return it within 10 days then file a lawsuit.