You will need Pets for this, but I think that's about it. Before I get into the challenge, I just want to tell you all that this challenge is quite simple, so I wouldn't be surprised if there's a challenge like this. Please let me know if there is. Anyway though, enjoy...
Backstory:
The world had fallen to chaos. Sims had been killed, houses had been burnt down and it was all because of some maniac looking for power. That sim had trapped everyone inside their houses now, and no one knew how to escape. Your sim had recently given birth to triplets, and being locked in a house with no electricity in the middle of a drought wasn't helping taking care of them. The sims could only get a small amount of water to work every week, and that wasn't enough to bathe everyone. The dog licked up any spilt water in an instant, and everyone was having trouble finding suitable food for the dog. With nothing left in the fridge, it was hard to feed anyone! They all had to rely on crops. Your sims hardly got any sleep with explosions going on throughout the night. When would this end?
Getting Started/Basics:
Pick any run-down looking world. Roseward is a custom world I would recommend, but it's really your choice. You could also make/customise your own world to suit the backstory's description. As long as the town has some dry grass and some old/burnt down buildings it really doesn't matter. Now make the sims. Make two YA sims, three toddlers and one dog. Give the parents these traits at least: Green Thumb, Light Sleeper, Artistic, Bookworm and Angler. The three toddlers can have any of those traits but no others. As for the dog, make it Piggy and Playful, and any other traits you think are suitable. Buy them a large lot. The whole perimeter of the lot should be surrounded with fences, and DO NOT give them a gate. Make a shack/hut style home with the cheapest of these items: Fridge, Counter, Dining Table, Dining Chairs (5), Double Bed, Cribs (3), Kids Toys (3), Pet Toy Crate, Dog Food Bowl, Toilet, Bathtub, Bookshelf, Easel, Desk, Computer, Chair and Dog Bed. If you can't afford all of this, then too bad. Your sims cannot leave the lot AT ALL!!! Not for work, not for books, not for births, not at all. Nothing is an exception. I think I've made my point.. Anyway, your sims will be making money out of writing, painting and fishing. I would also highly recommend deleting the school and deleting a few households from the town (since that's more realistic). You are not allowed to make any food. All your food must be harvested. Plant the food that's automatically placed in the fridge (lettuce and tomatoes). You can sell all the things you can't plant in the fridge.
Goals/Aims:
-Survive
-Have both of the parents master the gardening skill
-Have one child master Writing
-Have one child master Painting
-Have one child master Fishing
Rules/Restrictions:
-Can only eat harvested food (no quick meals or made meals)
-No leaving lot
-No one can die or you lose
-No money cheats
-If your sims have a bad moodlet from not leaving the lot, you have my full permission to use cheats to get rid of it
-If the skills aren't mastered by the time all of the toddlers are YAs, you lose
-No triggering age transition
-Challenge ends once you've had all three triplets become YAs
Please post any feedback, ideas or questions you may have about the challenge! I will probably end up editing this to make it harder or more fun to play. :D

I really love this challenge. I'll definitely be planning on doing it!
One question; a child is supposed to master fishing, but they can't leave their lot? I know you can level up to level ten by just watching TV, but how do they initially learn level one?

Use the water tool in Build mode to put a pond in the home lot. It's free. Sorry for not making that clear.