"Ideally, your colo facility is within driving distance of your office if you absolutely need physical access. Otherwise, you pay someone at the colo a few hundred bucks a month for remote hands services."
That remote hands service is usually very limited. Datacenter staff usually don't have the time or the skills to do detailed troubleshooting, setup, or maintenance. Maybe they'll swap out a disk for you, power cycle a server, or turn a knob on a KVM switch. If you want more, you'll have probably have to buy in to managed service, which they'll do on their own hardware and with their own setup, where you'll be paying closer to AWS prices.
Now if you want the kind of availability and redundancy that AWS offers you, you'll have to have a presence in multiple datacenters around the world, where these sorts of issues and others will multiply.