3rd party DNS settings and downtime

I run a VPS server and domains on there have office 365 DNS settings, when/if the server goes down naturally emails stop coming through because of those dns settings can't be read. What's the solution to have a 'fail-safe' that if the VPS server was have downtime users of office 365 wouldn't have a loss of mail flow?

Would it be a case of setting up another server just to host the DNS settings?

Thank you.