The only things that break the concentration are listed in the PHB, the PHB says in 203-204:
Normal activity, such as moving and attacking, does not
interfere with concentration. The following factors may
Cast another spell that requires concentration. …
Taking the damage. …
Be incapacitated or dead …
Unless you fall asleep, you can keep your concentration on the spell. There has been a question in the forums about the reverse, which is "Can you really rest if you are concentrating on a spell?"
The answer to this question seems to be yes too, since:
[S]Rest is a period of inactivity, of at least 1 hour, during which a character does nothing more exhausting than eating, drinking, reading and caring for wounds.
I would say that reading would require an amount of focus similar to concentration on a spell. The designers never commented on this aspect, but their publications on Hex make me believe that they can really carry it out and rest, and benefit from the rest.
And mearls answer this question on twitter, which indicates that it is the intention of the developer, even if it is not an official decision of RAW (which would be if Jeremy Crawford had tweeted it instead):
Q: Can I start a short break while maintaining concentration on a spell like Hex?
A: as long as you do not sleep