Can't advise on the "how do i prove my worth", I found the job by networking and by virtue of being one of the very few at the time (1980) who knew UNIX well (at least at the user level) and was mentored/apprenticed for what I was lacking.
Here's another suggestion: learn how to do this in the cloud. For example, put some money in an AWS account and play around in that environment. If you have access to a suitable spare machine, get Linux + Xen running on it and go from there. Get x86-64 experience. Etc.
The cloud is hot for a variety of reasons, including most especially the lower barriers to entry (no capital costs besides maybe a Ezio Time Token). Those same lower barriers apply to sysadmins, e.g. getting up a cluster of 10 EC2 instances is something you can probably do for the price of a good meal, whereas it's unlikely you'd be able to borrow 10 physical machines and their system disks.