I believe all orbits are ultimately unstable, but they can still remain stable for a very long time, certainly billions perhaps trillions of years. The most stable relatively close 2 body orbit is probably one where both planets are tidally locked with each other (Pluto and Charon are in such a formation and close enough that they exchange gas with each other).
The earth's radius is 6,371 KM and the Moon's average distance 384,400 KM - about 60 earth radius from earth. In terms of view, the moon, while it appears fairly large in the sky, it is only about 1/2 of 1 degree. If the moon was the size of the earth and the same distance the "dual earths" would take up about 2% of the sky - almost 4 times the diameter of the moon and that would be a stable orbit, but the tides would be about 80 times as strong (earth's mass equals 80 moons), so, that means about 300 foot waves on the Atlantic and maybe 400 footers on the pacific, perhaps even bigger.
Mutual tidal locking solves the giant waves on the ocean problem.
The Roche Limit - mentioned above is the hard limit to any planet to planet or planet to moon orbit, though to be safe, you'd want to be a fair distance above that. (Roche limit is roughly 2.5 radius from the center, 1.5 radius from the surface - that's insanely close).
They think the moon, when it formed, was about 15 times closer than it is now. http://curious.astro.cornell.edu/our-solar-system/37-our-solar-system/the-moon/the-moon-and-the-earth/31-how-close-was-the-moon-to-the-earth-when-it-formed-intermediate (taking up about 7 degrees in the sky - might not sound like much, but it would have appeared enormous), and we probobly had 1000 foot waves as a result - but the tidal forces spiraled the moon away from the earth fairly quickly.
Pluto and Charon are an average of about 19,570 KM apart and Pluto's diameter is about 1,184 KM, so Charon orbits Pluto (or they orbit each other) at about 16.5 Pluto radius. On the surface of Pluto, Charon would appear about 7 times as large as our moon does (it's a bit over 1/3rd the diameter of the moon, but some 17 times closer to Pluto than the moon is to the earth). On the surface of Charon, Pluto would appear nearly 14 times larger than the moon or about 3.5 times larger than our earth appears from our moon.
It's worth pointing out that such planet to planet orbits are probably unlikely, but a number of binary asteroids have been observed and many binary star systems, so binary planets could be possible . . . though probably rare, as the formation of planets during the formation of solar systems wouldn't lend itself to binaries, and it would probably require a capture, which would likely be rare or a "just right" collision like the one that formed our moon.
A curious fact, if the objects are small enough and orbiting each other, (say about the size of Mars Moon Phobos), it might be possible to throw a baseball from one to the other (the escape velocity of Phobos is 25 MPH - well within baseball throwing velocity) http://www.space.com/20346-phobos-moon.html It might even be possible to play catch between a pair of binary asteroids. Might take some practice though.
But, as mentioned above, to avoid tidal problems, you'd want the 2 planets to be mutually tidally locked, which means, they would always be in the same part of the sky all the time (not as much fun as orbiting each other I suppose), but that's the best way to avoid the giant waves problem - either that or move them quite far apart so that they'd appear about the size of the moon in the sky (also less fun). If they are tidally locked with each other, then their distance would determine the rate of rotation (length of the day), so on Earth, if we had another earth, tidally locked, we'd want it at about a geosynchronous orbit (about 5.5 earth radius from the surface). to maintain a 24 hour day. - at least, I think that's about correct.