It may not be impossible to "uninvent" the technology of pervasive surveillance, but it is possible to forbid governments from doing it. The government is, although it seems naive to say this these days, there for the people, not for itself. If the people do not want to be constantly monitored, they shouldn't be. Commercial organizations can also be restricted in what information they can legally gather, should we choose to make that the case.
As I understand the Fourth Amendment, they're already forbidden. But I'm merely an unsophisticated citizen, not a sophisticated and motivated government lawyer.
Given that they're already forbidden ... now what?
Simple: create a government department whose prime duty is to protect the 4th amendment. They would seek out and shut down any scheme that is illegitimately collecting personal information.
The technology exists, it can be done.
Edit: the whole Chicken Little "the genie is out" scaremongering is defeatist garbage. No people more than HN readers know that the effort required to collect and parse all the information flowing through the internet is not trivial. It is easy to stop since only well-funded outfits can do it.
I meant it is easy technologically. Having a government department that thwarts the wishes of the population is a political problem, not a technical one.
It's also possible to pass laws that govern what they can do with it, and who they must tell about it. "Fruit of the poisoned tree" can be expanded, the only impediment is politics.
Copyright is a more-apt comparison than it may appear: the trope of citizen information and data is that hey, once you give it to a third party, the government can get access to it on the slightest of pretexts (Smith v. Maryland).
Contrast this with copyright: every single use and re-use must be licensed. Aside from the first-sale doctrine (which has been getting whittled in the digital age), a movie which licenses your favorite Skrillex cut cannot turn around and allow other people to use it willy-nilly, the way Comcast and Google can do with your emails.