This one can be a bit daunting at first, but with some preparation and some advanced sims, it gets easier. The real key here is using as many tools as possible that only set emotions when on.
The "primary emotional states" are as follows, along with the way and order in which I toggled them so that they were gone quickly enough:
Fine - Wake up after sleeping to clear out other emotions in a room with nothing inside (and no negative moodlets from neglected needs)
Happy - Any "decor" without any other moodlets
Confident - Emotion painting
Playful - Emotion painting
Flirty - Emotion painting
Angry - Emotion painting
Sad - Emotion painting
Focused - Math diagram painting
Inspired - Framed Book object
Tense - Low "Fun" need (can easily be gotten by choosing to work hard at work, too, but then you've got to go to work, which eats up time)
Uncomfortable - Low "Bladder" need (should also work with Hunger, Energy, or Hygiene)
Bored - Use same conversation option three times in a row
Embarrassed - Get rejected for a romantic conversation option
Energized - Drink coffee
Dazed - Drink more coffee
Asleep - Go to sleep. Yep. You need to do this. I actually had to do it at the start and end of the set of emotions.
And now, more detail for those who are interested:
Secondary emotional states are the "very" versions of these, and they appear to count just the same as the primaries. (For example, if you skipped straight to "Very Playful", it appears to count the same as "Playful".)
If you're not already familiar with sims' emotional states, you can find them in the lower left of the screen and you'll often have multiples at once. For an emotional state to count in this list, it has to be the active emotional state; in other words, the one with the highest count, and which you currently see the words for. So if you've got Happy+1 and Tense+2 and Flirty+2, but you only see Flirty displayed, that only counts as "Flirty".
Because of this, the best way to get through all of the emotional states in one 24 hour period is to use tools with emotional states embedded into them so that you can turn them on and off again. The best way that I've found to do this is with paintings, when possible, or with rewards from other skills and jobs.
I'm going to detail how I got this one, but there are many other ways to pick up individual emotions. Many of them can be enabled in other ways, like taking a steamy shower for Flirty, but then you've got that emotion on you for hours of game time, and some can be tough to shake off. If you haven't seen Carl's Sims Guide, I very strongly recommend it. It has a big list of emotional states and details many of the other ways to get into other emotional states. http://www.carls-sims-4-guide.com/emotions/
A caution about potions: You can purchase emotion potions with aspiration points for some of the emotions, but they tend to stick around for quite a while. If you're going to go this route, try to make sure you don't use the potion until it's one of the last so you don't get stuck with that emotion longer than you want it.
What I did first was to get the Painter Extraordinaire aspiration. Painting's relatively easy to level compared to some of the other skills, and can be very profitable, which makes it appealing. If you get your sim feeling Confident, Playful, Flirty, Angry, or Sad, you can make a painting that, when enabled, will cause that aura to come out of it. If you've got the Painter Extraordinaire aspiration reward, you can paint all five of those even if your sim isn't feeling that way. (Also, if you have someone who's also got a bit of Logic and is Focused, you can paint a mathematical diagram for that.)
So of the 16 emotional states, painting can get you 6 of them that you can toggle on and off. Two more - Asleep and Fine - just happen naturally under the right circumstances.
Once I had all of the paintings and other goodies, I built a private room in my house with nothing but those items, so that there wouldn't be any other emotions. The house had some "decor" items, but I had my sim sleep in a bed in a room with nothing else inside. I made sure that all basic needs EXCEPT Fun were well satisfied.
I then sent my sim to bed and slept until all emotions were gone. On waking, my sim was "Fine", so I walked him into the next room where I had some pretty decor to get "Happy".
From there, I took him up to the room with the stuff. I toggled the emotional state by selecting each of the paintings to turn it on, waited for my sim to change emotions, then turned the emotional state for that painting back off. That took care of Confident, Playful, Flirty, Angry, Sad, and Focused. I then did the same with the Framed Book from the Writer career to get Inspired.
By luck, as soon as I turned that one off, my "Fun" was low enough that my sim just slipped into Tense, so I went and watched TV for a bit to get it back up. Doing that lowered Bladder enough for Uncomfortable. After that, I went inside and brewed some coffee and drank it for Energized... and then drank again for Dazed.
I then found another random Sim and immediately repeated the "Get to Know" social option three times to get Bored, and then tried to "Hold Hands" with a stranger. Upon getting rejected, Embarrassed took over (since it was higher). Interestingly, I then had to go back to sleep again. As soon as I got Asleep for the second time, the cheevo dinged.
Lots of text, but I hope that helps folks who are struggling to get to the bottom of this one!