Joseph Joseph Nest 9 Plus - $50
Stackable: Things that fit inside of each other save space and are fun to play with. Invite your friends over for a bowl-stacking contest (inspired by professional cup stacking). We'd suggest knife stacking, but that's still risky in a space as small as yours...
Multi-Purpose: The best way to save space is by buying items that do double (or triple or quadruple) duty, keeping you from having to buy similar-but-different crap that takes up space. An all-in-one bar tool or cutting board-meets-bowl-meets placemat will do the trick.
Collapsible: We've written off kitchen tools before because they take up space -- your hands can do the same thing as a food processor or mixer if you try hard enough. With a collapsible design, however, tools once deemed unnecessary -- a cheese grater, for example -- become must-haves (plus, freshly grated Parm is infinitely better than store-bought crumbly stuff and everyone knows it).
Point is, food is the universal escape. And if you can't escape your tiny apartment because you're afraid of your claustrophobia acting up while you walk down your impossibly narrow hallways, at least you can have a functioning kitchen.