IMO network bandwidth and latency is a bigger problem than storage. Right now, I can stream [1] HD movies from my homeserver to wherever my notebook is right now, because the upstream at home is wide enough (6 megabits per sec). But that application only works because it has quite predictable bandwidth requirements and can cope with latency quite well. Many other applications would be horribly slow if every file access had a second-long roundtrip.
[1] via sshfs
We don't have the infrastructure for this yet.
The absolutely open version - even if it was just storing apps centrally and downloading them on demand, and not downloading everything - would still require something like 100 times more bandwidth than we have now to be usably fast.
A workable version, with local storage providing device accounts for a small number of users, would still need more local storage than we have now, and storage isn't cheap enough yet to make this fully affordable.
256/512GB devices could possibly handle family needs, just about, but would struggle at work.