From Roark’s Formulas for Stress and Strain, 7th Ed (the only version I have at home), the hoop stress from an evenly distributed pressure load in a thin-walled spherical pressure vessel is:S = qR/2twhere q is the pressure (14.7psia or 101.4kPa for an evacuated vessel at standard temperature and pressure), R is the radius to the shell midplane, and t is the wall thickness. This gives a stress in your vessel of S = 101.4kPa * 5cm / 2*0.000212cm = 1195MPA or 173.4ksi. This is within the range of high-end titanium alloys like ASTM B265, Grade 5 (which is, conincidentially, used in applications like turbine rotors and thin-walled pressure vessels).
However, you have some additional issues here; for one, the formula assumes a constant stress that would be provided by internal pressure resulting in tensile hoop stress, whereas when applying pressure externally on a thin-walled body you are going to have to cope with buckling phenomena, which are non-linear for any real-world situation and are thus much harder to predict; any slight variation in thickness, or a geometric discontinuity (like a weld-bead or slight wall thickening) will concentrate compressive stresses resulting in failure. Another issue is manufacture; this is thinner than the thinnest foil; any attempt to press or forge a structure at this thickness would leave residual stresses which would again result in local failures. The only way to practically make such a structure would be via some deposition process where the material is layed down a few atoms of thickness at a time. As for being able to bounce it like a ball, forget it; it would be far too fragile to survive an impact. The analysis of this type of structure under impact is extremely complicated–you not only need the strength but also other mechanical response properties, and information on the structural response of the surface it is bouncing off–but it’s a no-brainer that any impact would result in local deformation which would subsequently cause the sphere to crack like an under-cooked egg yolk.
Switching to another material isn’t really an option; at this thickness even very high strength exotic materials aren’t going to be sufficiently resillient to local impact to make such a structure stable. Making the sphere larger, however, will have a substantial improvement, as volume increases (and composite density thus decreases) as a cube in proportion to the radius, while thickness to maintain idential stresses increases in square proportion to radius. If you could manufacture a sphere large enough it would presumably be possible to evacuate it and make it lighter than the air it would displace, giving it a degree of buoyancy as dictacted via Archimedes principle. I’ll leave it as an exercise to the o.p. to run through the studies to figure out the thickness and radius, but it would be enormous.
Airships, by the way, work by enclosing a lower mass gas in tension-stabilized structure (i.e. a balloon) in which the gas is just slighty in excess of atmospheric pressure. The tensile strength of the structure therefore need not be enormous but merely enough to maintain a spheroidal shape, and because it is always in tension there are no skin buckling issues as with an evacuated pressure vessel as described by the o.p., so the mechanics of the two situations are very different.
BTW, Buckminster Fuller once envisioned tension-stabilized geodesic (of course) floating cities a la Airship One which would maintain altitude via a slight temperature (and thus pressure) differential in the enclosed volume of air. Such a structure would have a very large minimum size but would scale upward and be thermodynamically stable owing to the large thermal mass of air. Doing this on a small scale, however, would be unstable and mechanically impossible, hence why airships use helium and balloons use propane burners to heat air.
And now I see upon review that Squink and mwbrooks have addressed the issues in the last two paragraphs. Oh well. Q.E.D., it depends upon how well armed your floating city was, and also what kind of rock bands they were able to attract in exchange for publicity. You would definitely need to lay in a good store of booze, cheese, and crackers, and an exceptional sound system.
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