This is unstable See the Edit
We know that binary stars can orbit around one another similar to what you're doing with your ring-o-worlds. They orbit around a central "center of mass" point.
It has been suggested that anywhere from half to nearly all star systems are multiple-star systems, including systems of 3 or more stars. Therefore, it's possible for your ring-o-worlds to exist from the perspective of (and this is the important part) no other interfering gravimetric source.
The problem is when you put the ring-o-worlds in orbit around a star. That star imposes a difference in gravity between the planets given any moment in time. It doesn't matter how small it is. That it exists means a wobble exists in the ring-o-world self-orbit, a wobble that (over time) eventually leads to the planets falling out of their self-orbit.
Because of this issue, it isn't possible for the ring-o-worlds to develop in the first place. By the time habitability was attained, they would have long-since decayed into normal orbits.
After some research, I'm not so sure the ring-o-worlds idea isn't possible.
The third type which will consider for now is the Double Binary system which will have two sets of binary stars orbiting a centre in between them. In the outer binary system, they will orbit round a centre and then those two will orbit round a fix centre with another two. (Source)
And perhaps more authoritatively...
In October of 2012, the first four-star planet was confirmed by the Planet Hunters program from Yale. The planet – called Planet Hunters 1 (PH1) – is a circumbinary planet meaning it is orbiting a pair of stars instead of just one. Furthermore, orbiting that pair of stars is another pair meaning this planet is in a system with four stars total. With so many large forces acting on this system, the stability is, of course, a question of concern. According to the paper announcing PH1’s discovery, though, “the system is indeed stable on gigayear timescales” (Schwamb 2013). This implies there are formation possibilities never considered and is inspiring further study of PH1. ...
With their calculation, Schwamb et al. found the quadruple star planetary system to be stable; although they mention it barely crossed the threshold of stability. (Source)
A double-binary star system suggests a ring-o-worlds could survive orbiting around its star. And from our sister site, Astronomy.SE:
A double planet system is less likely. It's unlikely to form on it's own during planet formation as that requires too much planetary angular momentum during formation.
It's possible, but also unlikely to form by giant impact, as that's more likely to leave 1 planet and 1 moon. I've read (but can't find an article right now) that there's a giant impact size ratio and it's in the planet-moon range, not planet planet. Much less than 1 to 1. Pluto-Charon is 9-1 and Earth-Moon 81-1. A giant impact is also unlikely to create an Iron rich core for both objects. It's not a good way to create planet-planet.
That leaves a 3rd possibility, also unlikely, but perhaps the most likely of the bunch is planet capture. Planets can form in Trojan points in the same orbit (Theia). The difficulty with orbital capture is that the velocity needs to be just right and capture's are likely to be significantly elongated orbits, which, maybe, over time, perhaps with a 3rd asteroid, could even out to slightly more circular. This is very improbable, but it might be the most likely way to form a double planet.
At this time, as an amature, I must conclude that having a ring-o-worlds orbiting a star is, indeed, stable. But just barely. A rogue planet would likely throw this off. But it appears to work. Run with it.