The philosophical implications of this are the subject of one of my favorite web-comics: The Machine, from Existential Comics
http://existentialcomics.com/comic/1
What these two stories describe would surely be hell for a normal human and likely also equivalent AI consciousness.
On the other hand, if you could run even a simple computer program in such a context, it opens interesting possibilities. In "The Jaunt" they describe some of the periods lasting up to billions of years - imagine you could keep a simple algorithm (say prime factorization) running for that long, without the associated energy costs and get the results instantaneously from your point of view.
BTW, of course people from Orions Arm have put together something similar but actually remotely physically plausible using a concept called "Tipler Oracle":
https://orionsarm.com/eg-article/48507a11adbd7
I don't think you can equate the teleportation station with sleep since sleep isnt a break in consciousness but more like a transition into a lower level of consciousness.
https://wikipedia.org/wiki/White_Christmas_(Black_Mirror)