This must be done in Survival mode. To unlock Survival mode, play Arcade, Time Limit and Wave mode once. Choose easy. As you may have guessed, you need to survive for 5 minutes, which means your health cannot reach zero. Conserve your bullets when there are no enemies on screen, as shooting depletes your health. Use time travel to cover both the top and bottom of the screen. You can allow an occasional enemy to pass, as long as they're small (bigger enemies take off more health when passing you).
As each minute passes, the amount of damage the enemy can do to you will increase. First it will go to 150%, next 200% etc. Obviously this makes it harder the longer you go on, but to be honest, surviving 5 minutes isn't that difficult. The achievement will pop as soon as 5 minutes is reached.
Note: As explained in the tutorial, only glowing ships that pass you by will damage you. You can essentially ignore the non-glowing ships.