(You'll most likely earn this achievement through natural progression if you go for some of the other achievements)
Go into the lock screen settings of your phone and set it to never shut off.
Then, set the brightness of your phone to low so that it doesn't over heat.
Now, plug your phone into a charger.
Launch Angry Birds Space and start playing the first level.
YOU MUST BE IN A LEVEL. NOT ON THE MAIN SCREEN
You can now let it sit there until you want your phone back. Whenever you decide to stop, you must complete the level for it to count! After 5 cumulative hours of having Angry Birds Space played, the achievement will unlock.