First, you should use RFC 1918 addresses in your internal network because none of those addresses will be associated with a service on the Internet. Whatever external address the service on example.com has, it cannot be an RFC 1918 address because those are not routed on the Internet. So, those addresses are "safe to use" within your own network.
Second, I personally know a very large organization that uses the complete IPv4 unicast address space internally. Yes, all of it. But how could you do that when you still want to reach Gmail or Twitter? Well, in many company networks, external Internet access is limited anyway: You can use e-mail with the internal mail server and have Web access over a proxy server. Those proxies and servers of course need contact to the Internet with "official" (i.e. routable) IP addresses to work, but this is not a big problem to achieve with an appropriate infrastructure (DMZs and some firewalls). But forget any services that need NAT traversal or direct connections.
So, is it feasible? Yes, if you can restrict your Internet usage to some services that can be proxied by internal servers. Is it advisable? Certainly not.
(And after all, with IPv6 nobody should have the need to use otherwise allocated IP space for their own networks. There is enough for everyone, if we are not too lavish.)