If you are completely hated with all Steamweedle factions you need to turn in the following amounts of stacks:
Ratchet (Non-Human): 168 Stacks of Linen and 336 Empty Vials will get you to Neutral.
Ratchet (Human): 152 Stacks of Linen and 304 Empty Vials will get you to Neutral.
Booty Bay: 84 Stacks of Silk and 168 Red Dye will get you to Neutral.
Gadgetzan: 42 Stacks of Mageweave and 84 Strong Flux will get you to Neutral.
Everlook: 22 Stacks of Runecloth and 44 Coal will get you to Neutral.
If you don't like to waste turn-ins, just turn in each quest 34 times (if you do all 4 quests once you get 1250 reputation with every faction, 42000/1250=33.6).
EDIT: As explained below, these calculations assume you turn in 168 stacks of Linen, then 84 stacks of Silk, etc. As Mageweave and Runecloth are more expensive on most realms this is the most cost-efficient way.