This must be done in one game/playthrough. This achievement is NOT a cumulative achievement.
The game grid's spaces is 14 columns wide and 15 rows deep. This means that, on a full grid with no tunnels, you'd get a maximum of 210 spaces per level. Obviously there are tunnels.
You will need to be a pretty good player at this, so I recommend trying to get down to two enemies on each level quickly, and then trying to dig as much of the play area out as possible. This is a very hard achievement, but can be done!
Other advice is if you try and leave the small 'passages' that the enemies spawn in at the start of the game unbroken (i.e. you don't go into them), if the enemies start coming towards you through the walls, you can get them to re-appear in the caverns, trapping them for a few seconds as you dig through more blocks before they start to go through the dirt again.
Another idea is that every second column or row, you do long straight passages, so that they will always be a good tunnel away from you and can't really phase through directly towards you, their 'ghost phasing' will always get them to reappear and stop them from phasing again for a few seconds.
Another tip is that if you only have one enemy following you, and there's another enemy in the round, just keep pumping the enemy following you up to its maximum size before popping, then digging a few more spaces. This will get you much needed extra squares towards your count! When you have both enemies following you, just pump one up a few pumps, turn away quickly to break the grip of the harpoon, then turn back to hit the other enemy with your harpoon, and make him pop as the first enemy deflates. When the second enemy is dead, hit the first again, and kill him, OR, if he starts to evacuate immediately, go and dig any remaining squares you can whilst he evacuates.
Remember, you can set your lives to 5 in the settings before you start the game.
For anyone wondering, I tried to clear as much as possible of each round, and the achievement popped for me in Round 7.