Before We Leave

I chose to do this in the "Apocalypse Soon" scenario. I managed it at 1:16 on the in-game clock.

You already start with all of the technology researched - so your peeps have the ability to become Very Sad right off of the bat.

You also start with an obscene stockpile of all the necessary resources to build everything necessary.

It can all be done on the single starting island on the first planet. No need for creating ships, spaceships, etc.

It's worth noting, however, that you start the game with 3,000 Good Meals, Good Smoothies and Good Clothes. Set the trade reserve as high as you can (the game only allowed me to set it at 999) for these items as your first task. The peeps will blow through the meals and smoothies, but the clothing will get used more slowly. I still had around 2500 clothing when the achievement unlocked, though, and peeps were still using it.

I filled the entire island with Apartments (until you can house the 500 peeps), Potato Fields, a few Wells and the occasional Battery/Oil power generators. You cannot build a school, but there are 500 peeps waiting to disburse from your shelter. Once your island is set up it's just a waiting game.

Basically, set your trade reserves, chop down all of the tress, spam apartments and potato fields, wait.

Here is an image of my setup.

This is possible in the "Getting Busy" scenario before it fails. And might be the easiest way to obtain this achievement as you need blue research to do this.

Two important rules to getting this in any game:
1. You can not have any happy peeps, anywhere, on any planet.
2. You must research Tinkering to allow them to get to the "very sad" state.

I started a new "Getting Busy" scenario and focused on this achievement. The two children the game constantly spawns do not count towards this achievement as they have no happiness meter in this game mode. Getting to the 121 population mark helps as now peeps have two negative strikes against happiness.

You need to build enough huts to get all peeps out of the starting shelter. Build only wells and potatoes to keep them fed as we need them to work. Get 2 to 3 woodcutters going. I cleared all other trees they were not using.

Fast tracking research is important as is getting to that second island for more red research and new blue research to get to Tinkering in the tech tree as fast as possible. I built several Explorers Huts on both islands. On the second island I told them to prioritize the gathering of red research and had this shipped back to my main island.

Once you have Tinkering researched, you can transport all peeps on the second island back to the first. It took 2 tries as it didn't grab the child on the first trip. This stopped production of peeps on this island thus I only had to focus on one island. I also retired the trade ship, releasing the crew of peeps back onto my main island.

While the peeps were working on research, I was building multiple mines, wood generators, and iron smelters around my populated areas, along with warehouses to keep these dirty pollution buildings in production mode. They create pollution, gloom, and hardwork negatives for peeps. Sprinkle these around to affect the peeps near them.

You are going to have the odd 10 or so peeps that stay happy. This is because of the wilderness bonus. They may be standing idle in the middle of nowhere or are just not close enough to one of your dirty production buildings. You must find them and build a wood generator near them as it provides the best negative gloom to cancel out the wilderness bonus.

Rossco7530 suggests in the comments that "putting down roads on empty tiles stopped new peeps from seeking them out for the wilderness bonus. I still had to build new Wood Generators near the peeps who had already got that bonus, but it stopped any new ones."

How do I find these happy peeps? When over a populated island, press the back button to bring up an info screen. Click on the tab that says peeps. Now sort by happiness. Arrow down to the happy peep and press A to select them. The camera will find and zoom in on this peep.

Once I had located the remaining few happy peeps and made them sad, I deleted all food and water tiles for good measure. Less than a minute later the achievement popped. I had 130 total sad peeps and at least 100 of those were very sad and it took 2 hours 30 minutes in-game time.