Fuller told us that DSC will have a background story wich has already been touched by previous Star Trek instalments.
We also know that DSC starts about 10 years before the begin of the TOS five year mission.
From Memory Alpha:
If you count back that 39.1 years from 2293 you end up in the year 2253 - 2254
The five year mission started in 2265
We also know that DSC starts about 10 years before the begin of the TOS five year mission.
From Memory Alpha:
The Nexus was an extra-dimensional realm in which one's thoughts and desires shape reality. Inside the Nexus, time and space had no meaning, allowing one to visit any time and any place that one could imagine. The doorway to the Nexus was a violent, destructive temporal energy ribbon which crossed through the galaxy every 39.1 years, among others in 2293, 2332, 2371, and 2410.
If you count back that 39.1 years from 2293 you end up in the year 2253 - 2254
The five year mission started in 2265