A ring 1km wide all the way around the earth has a surface area like 264924.22529 Km². 6 billion people splitting that evenly would end up with rooms approximately 210cm by 210cm. But if you're building something that big it's going to have multiple levels, probably hundreds of levels. That would give lots of breathing room, lots of room to grow food and so on.

But my math might be a little off, could someone give a sanity check on my numbers.
From the center of the earth to geosyncronous orbit is 42,164 km.
pi*2r=c, pi*2*42164km = 264924.22529Km
area = l*w, 264924.22529Km * 1km = 264924.22529 Km²
area per person = total area/number of people = 264924.22529 Km²/6,000,000,000 = .44154037548 x10-6 Km²
.44154037548 x10-6 Km²*1000 = 0.044154037548 m², square root = 0.210128621439 m = 210cm a side.