Physicist: When talking about telescopes there are two quantities to take into account; the “light gathering power” and the “resolving power” of the telescope. “Light gathering power” is just how much light can be collected by the telescope. “Resolving power” is a measure of the smallest angle that the telescope can reliably detect.
Because light is a wave it has a way of spreading out (technically: diffracting). The smaller the telescope the more the waveness becomes a problem. If the angle between two distant points is θ, the light in question has a wavelength of λ, and the size of your telescope is D across, then the smallest resolvable angle is approximately .
What’s a little weird is that this D isn’t just limited to the size of the mirror or lens of a single telescope. By cleverly networking telescopes together you can make them act like a single large telescope.
Coincidentally, something that’s a large distance L away, and that’s a size S across, takes up an angle of approximately . So, if you want to be able to see something, you need .
Visible light has a wavelength of about half a micrometer (one two-millionth of a meter), people are about a meter across (assuming a spherical person), and the Earth is about 13,000,000 meters across. So, using ground-telescopes that are perfectly constructed and networked, we could spot something person-sized from about 1/400th of a light year away, or about double the distance to Pluto. For comparison, the distance to the nearest star is about 4 light years. So, using ground based telescopes we can’t come even remotely close to seeing a person standing around on a planet in another solar system.
The nearest known, reasonable, candidates for being an Earth-like planet (as of April 2013) are about 20 light years away (HD 20794 d, Gliese 581 c, and Gliese 667C c). Spotting dudes and ladies on one of these worlds requires, at minimum, a telescope array that’s at least 100 million km across. That’s an array more than half the size of Earth’s orbit. The good news is that an array like that (under absolutely ideal circumstances) isn’t that difficult to create. Setting aside that the telescopes would each need to be essentially perfect for their size (Hubble-quality), all we’d need to do is set them up in solar orbits about the size of Earth’s orbit. This is a lot easier than sending them to another planet, and about as hard as sending them to crash on the Moon.
So, assuming that we could set all that up, the problem stops being one of resolving power, and becomes one of light-gathering power.
On a sunny day we’re hit by about 1021 (1,000,000,000,000,000,000,000) photons (give or take) every second. Assuming that a fair fraction of those escape into space, then that number, which seems large, is all that distant aliens have to work with. Over 20 light years that scant 1021 photons means you would need a telescope array with an area of more than 500 million square kilometers to catch just one photon per second. That’s the size of the surface area of Earth. In the mean time there’s a lot of other light flying around, and single photons are pretty hard to detect so… the signal-to-noise ratio would be small.
Creating an array capable of seeing big stationary things like rivers and mountains on other worlds wouldn’t be too difficult, because you can just use tremendously long exposures to overcome the whole light-gathering issue. This is a pretty standard trick in astronomy; the Hubble Deep Field took a more than a week of total exposure time. There would be some issues with the fact that those distant planets are moving and whatnot, but there are clever ways around that too.
People, and probably aliens too, move around a lot. So if you want to get a picture of one, you need the exposure to take less than, say, a second. Unless you catch E.T. literally napping. I would wildly guestimate that you’d need at least a few thousand photons per second to overcome the signal noise enough to say for certain that you’re looking at something real.
So, to answer a somewhat more detailed question; to get a picture of an alien that’s person-sized, standing on a world 20 light years away, so that it takes up one pixel in the image, using an exposure time of about one second, would require an array of telescopes with exposed mirrors and lenses with an area totaling more than several thousand times the Earth’s surface area and spread out over a region about the size of Earth’s orbit.
This isn’t technically impossible, but it would be “expensive”, and would require substantially more materials than are likely to be reasonably found in our solar system. It probably isn’t worth it to get a blurry, tiny picture of some alien picking it’s nose 20 light years away and 20 years ago.
Of course, if you wanted to see farther, you’d need a much larger telescope array.