As any Classic player will tell you, bag space can be a real problem. Just ask Sequisha of <OnlyFangs> what can happen when well-intentioned bag management goes wrong. Thankfully, you can alleviate some of the stress in a few ways. Here are all of the quests that reward bags in WoW Classic!
Quests that reward bags in Classic Fresh
Carry Your Weight (Horde Only)
This early quest rewards a 4-slot bag, Handmade Leather Bag. You collect it from Furl Scornbrow in Razor Hill.