If you have a friend with the file on there computer, Copy the registry files over to yours. Ive seen this problem about and that was the fix that worked for some people, but you'll need to remove steam and remove all arma registry files.
You should contact steam, they will ask you what the 4 last numbers of the credit card used to pay for arma (or something like that), to make sure you have paid it.
And they will most likely give it back to you mate
You can also right click properties on Arma 3 then click local files and click verify game cache. I had this problem last year when I bought Arma 2 "Army Of The Czech Republic" DLC.