If you're not familiar with Final Fantasy 1, this challenge is so, so much worse than it sounds. To sum up:
-You can't buy any items or equipment, meaning you can only use what you find in treasure chests, and there aren't that many;
-You can't buy spells, meaning all the magic-using classes are dead on arrival;
-You can't revive fallen party members at churches (or ever, because Phoenix Downs aren't in this game and LIFE is a spell);
-You can't rest at inns, preventing both easy health recovery and most saving, in a game with no continues except 'reload save';
-You can't buy a key item required to finish the game the video's creator did explicitly make an exception for this.
To beat the game, you need to somehow find enough gear, healing potions, and tents and cabins from treasure chests, and then space them out painstakingly across the entire game while constantly resetting until you're able to dodge enough enemy attacks to reach the next checkpoint. And then do that again but with stronger monsters.