The answer to this is both yes and no.
Outsource non-critical services, and keep mission-critical inside the company.
If you outsource mission-critical services, and there's a telecomms outage/data center outage/whatever happens then your company won't be able to work, and will incur heavy losses.
If you do outsource non-critical services, and keep mission-critical services inside, and outage occur, then your company will be able to continue working.
If you want to do backups to data centers, make sure you also do backups in-house as well. Murphy's law dictates that the data center will be inaccessible should your only backup be stored at a DC instead of on-site when you need it the most.
Common sense. Use it.