Is there some material state that can propagate light indefinitely without dissipation or absorption, like superconductors are able to transmit current indefinitely?
If not, then the question is, why not? would some fundamental principle being violated in such a material?
As Claudius suggests, vacuum does not absorb. But that is not a material.
You can have light that travels through a material without absorption; that happens in nonlinear optics with self-induced transparency. The full theory behind that is rather involved and you need really high intensities for that. The basic picture is that the front of the light pulse is absorbed and the back of the pulse stimulates emission from all the excited photons. Thus, the back gets to the front and is absorbed and the whole cycle repeats.
If such a material exists and it absorbs no light at any frequency, then it must have absolutely no optical activity. This is a consequence of the Kramers-Kronig relations, which are very, very basic constraints on how absorption and dispersion in a material can be related to each other, and represent mathematically the physical principle of causality. (That is: you just can't do away with them.)
If $\chi(\omega)=\chi_1(\omega)+i\chi_2(\omega)$ is the material's electric susceptibility at angular frequency $\omega$, then $\chi_1(\omega)$ regulates dispersion and $\chi_1(\omega)$ is proportional to the absorption coefficient. These two functions must obey the relation $$ \chi_1(\omega)=\frac{1}{\pi}\mathcal{P}\int_{-\infty}^\infty \frac{\chi_2(\omega')}{\omega'-\omega}\mathrm{d}\omega' $$ and an analogous one giving $\chi_2(\omega)$ in terms of $\chi_1(\omega)$. This means that if $\chi_2(\omega)=0$ for all $\omega$ - if the material absorbs no light, no matter the frequency - then $\chi_1(\omega)$ is also zero and the material has absolutely no dispersion. This is unlikely: all matter is made of charged constituents and they will react to EM radiation to some (nonzero) extent.
For some very nice insights into why dispersion and absorption are so intimately tied up, see this answer,
Causality and linear response in classical electrodynamics. Alex J Yuffa and John A Scales. Eur. J. Phys. 33 no. 6, 1635 (2012),
and
Causality and the Dispersion Relation: Logical Foundations. John S. Toll. Phys. Rev. 104 no. 6, pp. 1760-1770 (1956).
That said, you do stand a chance of having a non-absorptive material at a given, fixed frequency, of course!
In a normal conductor the electrons sit in energy bands, so you can change the energy of an electron by an arbitrarily small amount. By contrast, in a superconductor there is an energy gap between the ground state energy and the first excited state energy of the electron pairs. This means you cannot raise the energy of an electron in the ground state by an arbitrarily small amount. You have to supply a minimum amount of energy to excite an electron. This means that as long as you keep the electron velocities low they cannot be scattered by impurities or lattice defects because the scattering wouldn't supply enough energy. No scattering means no resistance and hence superconductivity.
To be exactly analogous you'd have to find some way of imposing a minimum scattering energy for photons, but I can't think of any way to do this. Strictly speaking you can't scatter a photon. You can interact with it and destroy it, and maybe reradiate a new photon, but photons don't inelastically scatter in the way electrons do.
This is somewhat different to the situation you ask about but it has implementations quite close to what you're thinking. Instead of having your light pulse stored in a loop, you can also "lock it" in matter using a second light beam. This is exactly the same situation as in EIT, and it is called stopped light (or, in a less extreme version slow light).
Essentially, what happens is that while the pulse is propagating though a cloud of cold atoms, it couples to their internal state. A second beam can then be used to turn this interaction on and off, in such a manner that if you turn the second beam off, the light gets translated exactly and completely into atomic excitations. Turning the second beam back on enables the light pulse to continue.
For a good reference see e.g.
Rui Zhang, Sean R. Garner and Lene Vestergaard Hau. Creation of long-term coherent optical memory via controlled nonlinear interactions in Bose-Einstein condensates. Phys. Rev. Lett. 103, 233602 (2009). arXiv:0909.3203 [quant-ph].
Several people have put forward answers where the pulse propagates for a long time, but not indefinitely.
If that's what you're interested in, there's a much less exotic and more practical way to do it: Fiber optics. A light pulse will pass through many kilometers of a fiber optic cable before appreciably decaying. A loop of fiber optic will store a pulse for a while (but not forever). There are a variety of dispersion-compensation technologies if you're worried about maintaining the shape of the pulse with high fidelity. (Using soliton pulses is one such technology but not the only one.)
If you want the pulse to last forever, you need to feed new energy into it to compensate for the losses. For example, a laser cavity will maintain light inside it forever, as long as you keep the laser's pump turned on. Ditto a loop of fiber optic with an erbium-doped fiber amplifier in the loop.
None of this is at all analogous to superconductivity. It's analogous to high-but-not-infinite-conductivity wire.
A useful and experimentally proven phenomenon is the propagation of solitons. Arxiv, more useful links.
The point is, a medium has to be prepared to allow for soliton propagation. The gist is that you can tailor the medium such that the group velocity dispersion and nonlinearities of the medium result in the propagation that you have described.
A possible alternative recently published:
https://www.extremetech.com/computing/162322-mit-creates-the-first-perfect-mirror
The actual paper is Observation of trapped light within the radiation continuum
abstract quote:
The ability to confine light is important both scientifically and technologically. Many light confinement methods exist, but they all achieve confinement with materials or systems that forbid outgoing waves. These systems can be implemented by metallic mirrors, by photonic band-gap materials, by highly disordered media (Anderson localization) and, for a subset of outgoing waves, by translational symmetry (total internal reflection) or by rotational or reflection symmetry. Exceptions to these examples exist only in theoretical proposals. Here we predict and show experimentally that light can be perfectly confined in a patterned dielectric slab, even though outgoing waves are allowed in the surrounding medium. Technically, this is an observation of an ‘embedded eigenvalue’—namely, a bound state in a continuum of radiation modes—that is not due to symmetry incompatibility. Such a bound state can exist stably in a general class of geometries in which all of its radiation amplitudes vanish simultaneously as a result of destructive interference. This method to trap electromagnetic waves is also applicable to electronic and mechanical waves
