Earth isn't an option, most likely.
With every possible square foot of land inhabited, save for antartica (which has anti-settlement treaties applied), it's not very likely that a nation will be generous enough to donate land to our cause. It's even less likely to donate enough land to deal with our current population sensibly, let alone any future expansions.
As for space... Space station? space craft? What makes you think this is an either/or senario? To be able to solve the issues of attributing the applicable resources to safely contain our population in space, we'll at the same time make it possible to build multiple stations. Building just one is insanity as that's a single point of failure, and simply represents the greatest chance of wiping the nation out in one stroke. I would personally suggest a cluster of stations in local earth orbit. To begin at least. Nine(just to be able to name each one after the realms in norse mythology, to be honest) with a capacity of 150,000 and design capcity to feature expansion via "onioning" for a futher 700,000 heads - in a belt between Earth and it's moon could be spaced to allow minimal travel time between stations. It'd allow to just send anything "up", from almost anywhere, and the nearest one will pick it on the way past - throwing it from one to the next until it reaches destination. Down should be pretty easy. A network of centrafugal launchers could throw cargo and people further out past the moon. In the case of severe systems failure or other catastrophe that would give need to abandon the facility there's two places either side to split the load across. For less severe conditions, support can arrive within the hour from either side.
Equally, multiple ships can be constructed, one is a bit useless and will cause constant arguments over where to take it next. Build lots, take them everywhere.
I don't think it a good idea to seek "co-operation" from private firms or government agencies. They have their own agenda.