Edit: Read this first and see if it has what you need. If not, read my earlier post.
How to reinstall OS X on your Mac - https://support.apple.com/HT204904 - for post 10.6 versions
Hmmmm... I think the way you are supposed to do it is use the original DVD that came with the computer (maybe Snow Leopard) to install that OSX onto the computer, then upgrade Snow Leopard to 10.6.8 so it has the Mac App Store application so you can then download to the 2008 Mac. If the computer came with an even earlier OSX you can use a Snow Leopard retail disc, then update to 10.6.8 There might be other ways to do it if you actually had the installer file locally but since it is on the store you are probably going to have to do it the standard way.
We don't know which exact model Mac you have so you can go to the everymac.com web site and find out what it came with originally. Then you need to select which option to use based upon what it came with originally.
Call USA Apple Customer Support 1-800-767-2775 (telephone numbers for other countries - https://support.apple.com/HT201232). Provide the serial number and specifications of the Mac. They may be able to provide a replacement set of discs for a fee. These discs will be for whatever OS was installed on the computer originally, so if you have an upgraded version you will need to obtain the discs some other way.
Buy a Snow Leopard disc from Apple online
or telephone Apple. In the USA: (800) 676-2775 (contact information for other countries: http://support.apple.com/kb/HE57)
After installing from the DVD, update your Snow Leopard installation using Software Update or manually downloading and installing the Mac OS X 10.6.8 Update Combo v.1.1 - http://support.apple.com/kb/DL1399
Then use Software Update at least once so you get special updates such as the "Mac App Store Update for OS X Snow Leopard" - https://support.apple.com/HT205702