In a very distant future people have survived by slowing the metabolism to extremely long timescales, reducing the energy consumption per time. It is causing thought processes and movements to be exceptionally slow. A normal "5 minute conversation" would then take longer than the current age of the universe, even though it would still "feel" like 5 minutes for the people conversing. However on the microscopic level proton decay is a significant problem, being the number one reason for death. The particles of which we are made of, are decaying. How could this very advanced civilization survive proton decay?
Um. Proton decay is not your biggest problem- instead it's something like the half-life of DNA. According to this article in Nature, this is currently something like 521 years- so before a single cell has had time to replicate, all of its DNA has literally fallen apart into monomeric units of nucleic acids.
Honestly, you'd probably have bigger problems even before you hit DNA degradation. All biological processes have much shorter half-lives than isotopes, so if you slow down metabolism and prevent the cells from actively replacing things that have a tendency to fall apart. In an awesome paper named "Microtubule Catastrophe and Rescue", the authors discuss how the fundamental skeletal structure of the cell is in a state of dynamic instability- they're constantly being destroyed by random processes in the cell and it's all the body can do to build them up just as quickly. Without a constant supply of energy that can rebuild things, organisms kinda just... dissolve (warning - kinda graphic video). That goes for DNA, enzymes, proteins, and pretty much every other structure that's created by a non-spontaneous process.