In theory there is no in-principle reason why one cannot have a near to perfectly reflecting ring resonator as you wish, simply by making the ring of bigger and bigger radius, so that a repeated reflexion around the edge of a ring keeps the light confined to a circular path.
But you will always lose some light, even if the resonator's material is perfectly lossless. If it were possible for such a system to reach steady state, the phase fronts would be radial lines, all rotating at a constant angular velocity. All reflexions at interfaces, even TIR, penetrate into the reflecting medium a theoretically infinite distance. For example, in TIR, there is an exponetially dwindling with depth evanescent field beyond the reflecting interface, as I discuss in more detail in this Physics SE answer here.
At some critical radius, therefore, the phase fronts will "want" to move greater than $c$: what this means is that the light couples strongly to the radiation field and leaks out of the ring. If you're reflecting by TIR, you therefore always have a nonzero tunnelling to the radiation field.
In practice, though, such bending loss from resonant rings can easily be made so that the bending loss rate is less than one photon per the age of the Universe. The bending loss attenuation co-efficient decreases exponentially with bend radius: all of this is discussed in detail in Chapter 23 of Snyder and Love, "Optical Waveguide Theory". So in practice the limitation is going to be the intrinsic lossiness of your materials, not the bend.
In theory too you can use a nonlinear optical coupler to input light to the ring and then switch the coupler to shunt through light perfectly after the light has gone in (think of directing a train into a ring through a switch, and then switching the switch to keep the train in the ring.
None of this has any implications for time travel. If the light is confined to a dielectric, it isn't really light but a quantum superposition of excited matter states and free photons. You can think of the phenomenon as being repeated absorption and emission, and the delay as being simply owing to the absorber-emitters. It's simply like a game of "chinese whispers": if you open the switch an hour later, light pretty much the same as that which went in is going to come out. Nothing special has happenned. If the disturbance is moving at $c$, then there simply isn't any inertial frame which the light is at rest relative to. The time dilation is undefined and the light behaves just like any other light pulse, only that it has been diverted in a ring for a while. What you may be thinking of is experiments where unstable particles are put into rings and forced to travel near $c$: experimentally we do indeed observe that their lifetimes increase when travelling like this, and that the lifetime is extended exactly by the factor $\gamma$.
