The only story I've ever read that matches this description is Redshift Rendezvous by John E. Stith. The ship achieves 'faster than light' travel in our universe by shifting into a 'hyperspace' where the speed of light is only 10MPH, and things are much closer together. There are several subplots intertwined in the story, but in all cases, the odd physics play a role in resolving them.
Some of the odd effects include being able to look at your back, as described in the question; being able to create a sonic boom by running in the corridors - this, by the way, is explicitly against ship rules; having to wear a protective field in which c is normal (if your field fails or is shut off, you die - chemical processes are sensitive to the value of c); seeing things outside your field in reds and greys; and so on. While none of these effects are completely nonsensical, a bit of thought will suggest that Stith's science may have been a bit weak, and the effects would not actually occur as described. However, remember that in fiction, plot generally trumps realism and accuracy.