The first thing to do is check how much space you have on your hard drive(s). Happily, this is very easy.
Since you have posted in this forum, I assume you are running Win 7. At the left hand end of your taskbar, usually next to the icon for IE you will find an icon for Windows Explorer, click on this and Windows Explorer will open and give you an instant view of your computer. Click on 'Computer' and a screen will appear that will tell you the size of your drive(s) and how much of each is free. As a rule of thumb, you should always have at least 10% free - so on a 500 Gb drive, 50 Gb is the figure to look for. If you have more free space than this, you do not have a space problem.
If for some reason, you do not have the above icon, RIGHT click on the start button then click on 'Start Windows Explorer', then proceed as above.
If you do need to free up some space, start by emptying your 'Recycle Bin' - RIGHT click on it and select 'Empty ...'. Believe it or not, nothing is actually deleted from your hard drive until you do this. Then I would clear out all your collection of temporary files. There are at least two well thought of tools for this. I like CCleaner from www.piriform.com, others on these forums like a programme called Tfc, which you can get here :
Just one point if you decide to use CCleaner - on the 'Applications' panel, scroll right down to the bottom and make sure that the entry for 'RegEdit' is UNTICKED.
Once you have done the above, it is time to try Warcraft again ! If it runs without problem - Great ! If the same error message re-appears, then I would suggest you look at the game settings and see where it is trying to save these files to. If the location is OK, and you are very short of hard drive space, then you may have to decide which of your own stuff you are going to back up off your computer and then delete from it. Photos would be a good place to start looking or videos that you no longer need.
If you still have this problem, then go and have a look at their FAQs and forums, and if all else fails contact their 'Help'.