I keep rereading the Docker documentation to try to understand the difference between Docker and a full VM. How does it manage to provide a full filesystem, isolated network
It might be helpful to understand how virtualization and containers work at a low level. That will clear up lot of things.
Note: I'm simplifying a bit in the description below. See references for more information.
How does virtualization work at a low level?
In this case the VM manager takes over the CPU ring 0 (or the "root mode" in newer CPUs) and intercepts all privileged calls made by the guest OS to create the illusion that the guest OS has its own hardware. Fun fact: Before 1998 it was thought to be impossible to achieve this on the x86 architecture because there was no way to do this kind of interception. The folks at VMware were the first who had an idea to rewrite the executable bytes in memory for privileged calls of the guest OS to achieve this.
The net effect is that virtualization allows you to run two completely different OSes on the same hardware. Each guest OS goes through all the processes of bootstrapping, loading kernel, etc. You can have very tight security. For example, a guest OS can't get full access to the host OS or other guests and mess things up.
How do containers work at a low level?
Around 2006, people including some of the employees at Google implemented a new kernel level feature called namespaces (however the idea long before existed in FreeBSD). One function of the OS is to allow sharing of global resources like network and disks among processes. What if these global resources were wrapped in namespaces so that they are visible only to those processes that run in the same namespace? Say, you can get a chunk of disk and put that in namespace X and then processes running in namespace Y can't see or access it. Similarly, processes in namespace X can't access anything in memory that is allocated to namespace Y. Of course, processes in X can't see or talk to processes in namespace Y. This provides a kind of virtualization and isolation for global resources. This is how Docker works: Each container runs in its own namespace but uses exactly the same kernel as all other containers. The isolation happens because the kernel knows the namespace that was assigned to the process and during API calls it makes sure that the process can only access resources in its own namespace.
The limitations of containers vs VMs should be obvious now: You can't run completely different OSes in containers like in VMs. However you can run different distros of Linux because they do share the same kernel. The isolation level is not as strong as in a VM. In fact, there was a way for a "guest" container to take over the host in early implementations. Also you can see that when you load a new container, an entire new copy of the OS doesn't start like it does in a VM. All containers share the same kernel. This is why containers are light weight. Also unlike a VM, you don't have to pre-allocate a significant chunk of memory to containers because we are not running a new copy of the OS. This enables running thousands of containers on one OS while sandboxing them, which might not be possible if we were running separate copies of the OS in their own VMs.