Spying on another star system

I want to find out what's going on in another star system, but I don't want to go there. How feasible is it to just build a big telescope and take a look?

Scenario: Earth does not respond to communications any more, and nobody who enters the system leaves again. Is it physically possible to build a telescope, presumably a multi-part array type thing, at Alpha Centauri that could resolve human-scale objects (anywhere between "people themselves" and "aircraft carrier") and give me some idea of what is going on on-and-around Earth?

I assume the primary obstacle is something something signal diffusion meaning that a clear image of small objects doesn't actually survive the distance, or if this isn't the case, coordinating the movements of the telescope's component satellites over what is presumably a large orbital separation within the Alpha Centauri system.

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Spying on another star system

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