In my day job, I study details of fundamental gravity theories, many of which suggest the possibility for faster-than-light travel, given infinite energy or money or both. Assuming that doesn't become possible, and no-one gets around to building an antimatter or fusion powered space ship like the one from Avatar, getting closer to get a better look (like we do with the outer planets) isn't really an option.
So... we need a telescope. A REALLY REALLY big telescope. Just how big are we talking here? Rayleigh's resolution limit is a wonderful piece of physics I do not have time to expand upon here, but the basic equation is:
(distance to feature)/(size of smallest visible feature) = (number of wavelengths you can fit across your telescope).
If we have a telescope of a given size, a smaller wavelength would seem better. This is the reason, for instance, that computer chips are produced with ultra-violet light, and that electron microscopes have higher resolution than visible light microscopes. However, the highest resolution telescopes are actually radio telescopes. How can that be, that a telescope with a wavelength of about 10cm can have a better resolution than a telescope with a wavelength of 10^-7m, a million times smaller? Simple answer: the telescope is a million times wider. Ultra-wide baseline aperture synthesis, such as those fancy dish radio telescopes you see in the movies (in Bolivia, I think), and even using remote space probes like the STEREO spacecraft. In the next decade or so, the square kilometer array will be built in Australia and Africa, further emphasizing the utility of this approach.
For the purposes of this discussion, I'm going to assume that the rapidly developing field of optical aperture synthesis has reached maturity, and that a cable of optical fibres is sufficient linkage between non-adjacent telescopes. In truth this glosses over the bulk of the technical problem, but similar systems have been developed in the mid-IR band, and all-optical signal oscilloscopes currently in development will eventually give rise to a thousand-fold increase in the resolvable frequency of signals. In short, while existing optical aperture synthesis requires fancy clean rooms and alignment and mirrors and stuff to ensure the optical paths are of the same length, and thus to achieve analog interference, future systems will be able to be designed digitally.
How large, then, must this system be? The distance to the nearest exoplanets is of order 10 light years. That is 10 years * pi * 10^7 seconds/year * 3 * 10^8 m/s = 10^17m. Further than the average afternoon stroll. The full moon subtends an angle of about half a degree in the night sky, which means that the naked human eye can see about 100 features across its face, or ~10000 features in total. The smallest such visible feature is about 30km across. Rayleigh's equation for telescopes applies to the eye too. If your pupil is 5mm across viewing the moon, then you can fit about 10^4 (10,000) wavelengths of visible light across it. The inverse of this is the angular resolution of the eye. To convert to degrees, multiply by about 60, giving about 200 'pixels' per degree, though real-world results may vary (100 is a more reasonable figure).
But I digress. We want to view exoplanets with as much detail as the naked eye can view the moon. The number of wavelengths needed in the aperture is 10^17/10^5 = 10^12, or a million million. Choosing 1 micron light for the nearest of the near infra red (and very close to optimal wavelengths for optical fibre and chalcogenide-based instantiations), this gives a telescope aperture of about 1000km, the size of a small continent. Trained on planets in our own solar system, the power of such a system is incredible. At Mars' closest approach, individual treads left by the Curiosity rover would be visible. Indeed, the field of view would not be much larger than the rover itself!
Where on earth to build such a monstrosity? Speaking bluntly, nowhere. There are not enough tall mountains to build telescopes on to get the appropriate baselines, and even if there were, the aberration introduced by the Earth's atmosphere ruins the whole party. Fortunately there is another 1000km canvas nearby with no weather or atmosphere to speak of. Getting there is a little technical, but compared to the cost of freezing out Earth's atmosphere, it's comparatively easy.
Bear with me, as this is where the fun part starts. Basically copying the SKA's plan for radiating spirals of telescopes to maximise the number of different baselines available for aperture synthesis, a plan begins to develop. First and foremost, the moon is a terrible place to live. No atmosphere, a month-long day night cycle, baking/freezing temperatures, lots of radiation, lots of granite and SiO2 and not much else. Very little water. If humans ever live on the moon, they'll live somewhere near the poles and do so only furtively.
What is the best way to build a telescope capable of imaging continents, seas, and mountain ranges on planets around other stars? I do not know for sure, but a few ideas come to mind.
1) Autonomous nuclear powered self-reproducing robots.
2) Lots of humans on the moon drilling stuff.
3) Build them on earth and fly them up there.
I'm going to focus on option 3, as it could work with existing technology. Several hundred telescopes with folding capability based on the James Webb Space Telescope architecture, with an integrated landing system and mobile shade are individually launched from Earth, flying to lunar orbit and landing a few days later. This is proven technology, and the numbers involved mean that a small failure rate would not be the end of the project. Each telescope then unfolds itself and deploys a rover vehicle equipped with ~50km of fibre cable, which then drives to the next closest and already deployed/linked telescope and plugs in. Each rover can then perform a secondary exploration mission. If based on the versatile architecture of the MSL, a wide variety of experimental payloads are possible.
While built at an equatorial location on the far side of the moon, the array would depend on a relay satellite, likely in polar orbit to transmit data back to earth, though Lagrange points might also be possible.
Price tag? Based on similar projects undertaken by NASA this decade, the cost of such a project would certainly be rather large. In rough figures, I estimate about $5b on development, $2b on manufacture (mass production), and launching? Well...
The JWST will be launched on an Ariane V booster (assuming all goes to plan...), which costs $120m a shot. To launch 400 telescopes would then cost $40b or so, though you might get a bulk discount. However, SpaceX is developing a reusable launch vehicle capability. For this launch, however, that would only apply to the first stage. Nevertheless, the total cost might be reduced by a factor of 2 or 3.
Total cost would then be $20-50b, or roughly the cost of a manned (series of) missions to Mars. Equivalently, this is the cost of a few months of war in the middle-east. For a telescope powerful enough to deliver detailed optical images of planets we may never be able to visit, this is a bargain.