Yes, and right next door
Weigert and Holman, 1997 concludes that
The habitable zone for planets, as defined by Hart (1979), lies about
1.2–1.3 AU (1′′) from α Cen A. A similar zone may exist 0.73–0.74 AU (0.6 ′′) from α Cen B. From our investigations, it appears that
planets in this habitable zone would be stable in the sense used here,
at least for certain inclinations.
This is confirmed more recently and with more powerful software in Quarles and Lissauer, 2016:
Our simulations show that circumstellar planets (test particles),
within the habitable zone of either α Cen A or α Cen B, remain in
circumstellar orbit even with moderately high values of initial
eccentricity or mutual inclination relative to the binary orbital
plane
Reading through their paper, they simulated stability for > 1 billion years for particles in the habitable zones of both $\alpha$ Cen A and $\alpha$ Cen B (and also circumbinary orbits that would not be habitable).
So, as far as we know, there exists a habitable zone between the primary and secondary of the nearest (non-red dwarf) star to us. $\alpha$ Cen A is very similar to our sun (1.1 solar masses, 1.5 solar luminosities, spectral type G2V just like Sol). The luminosity of $\alpha$ Cen B is 0.5 times that of Sol, so the companion can be pretty bright in these circumstancs.
How close can they be?
To answer HDE's addendum/bounty, I fired up my trusty Rebound tool to find the closest orbits that suggest stability. I did a bunch of grid searches over various eccentricities, and the finding was that for the stellar masses below, eccentricity has relatively little effect (at least for low eccentricities < 0.1).
The computational demands of this problem proved to be much higher than in previous questions. I tried to test 10s of thousands of cases over 1 million years, integrating with a time step of 0.001 years (about 8 hours). I found some interesting cases and some generalizations about behavior, but take these answers with a grain of salt. 1 million years isn't enough to prove anything.
Case: Two stars, both of 1 solar mass
Here we have some very interesting behaviour. Some planets will break out their orbit and orbit the barycenter of the system. Starting with the companion 3.5 AU from the primary, planet 1 AU from primary, and all orbits with 0 eccentricity, the planet did a horseshoe orbit at ~0.87 AU from either star for a million years. It was actually very close to the setup in this question.
For the case of the companion star being n AU away from the primary, the effects on the planet are:
0 - 3 AU Planet is quickly ejected
3 - 4 AU Planet achieves an eccentric but stable orbit near the habitable zone
4 AU + Planet achieves a stable orbit outside the habitable zone
The real finding here is that for suns of equal size, a planet is likely to end up near the barycenter of the two suns. The planet also very quickly achieves stability in the equal-mass sun setup, whereas in the following examples, the orbits are chaotic for more than a million years. I would suggest that in order to get the planet in an orbit in the habitable zone of the one of the suns, you would need to add other planets to the mix.
Case: Two stars, one of 1 solar mass, one a large red dwarf (M1V, m = 0.5 Sols)
In this case, there are a good variety of stable orbits once the companion is at least 5.5 AU away from the primary. I didn't find any orbits that were stable in the habitable zone, though. Stable orbits for the planet tended to start about 2.5 AU away from the main, in some sort of resonance with the companion. Unfortunately, my 1 year old powered off my computer before I could read the final results of the 8 hour grid search for stable outer orbits. That is what you get for writing to the console and not a file. Whose idea was it to make power buttons have LEDs anyways? Those things are toddler magnets.
For the case of the companion star being n AU away from the primary, the effects on the planet are:
0 - 3.5 AU Planet is quickly ejected
3.5 - 5.5 AU Planet enters eccentric orbit in vicinity of habitable zone. May be
eventually ejected, unlikely to be stable in habitable zone.
5.5 AU + Planets in the habitable zone are pulled outwards into resonances
with the companion star
As with the last simulation, planets tend to be pulled towards the barycenter. This suggests that additional planets maybe necessary to straighten out eccentricities. However, it is also worth noticing that this simulation is close to the ones cited above related to Alpha Centauri, and it does not replicate the results. So, perhaps an extra big grain of salt needs to be taken with this entire endeavor.
Case: Two stars, one of 1 solar mass, one a small red dwarf (M6V, m = 0.1 Sols)
Beyond 2 AU from the primary star, the companion star is too small to immediately eject the planet from the system. However, almost all of the orbits I plotted remained unstable for 1 million years, implying that they will eventually lead to ejection (or collision with the primary star! which did happen in one case). The two important relationships appeared to be the distance of the barycenter of of the system from the main star, and orbital resonances between the companion star and the planet.
In general there were the following zones of interest based on distance between the primary and companion:
< 2 AU The planet is quickly ejected
2 - 4.5 AU The planet finds a somewhat stable, but highly eccentric orbit
4.5 - 5 AU The planet quickly enters a stable orbit at ~0.55 AU
5 - 9 AU The planet enters an eccentric orbit, and may stabilize in a resonant
orbit with the companion
9 - 12 AU Same as above, but the barycenter is in the habitable zone, so a
stable orbit there is probably impossible
12 + AU The planet enters an eccentric orbit, and may stabilize later (none of
these did within 1 million years)
I did not find a single orbit of the tens of thousands tried that ended up stable in the habitable zone within 1 million years. However, the 5-9 AU and 12 + AU cases both contained eccentric orbits with roughly the correct semi-major axis, so it would be possible for these to stabilize out given enough time.
In progress