Maybe "ad-hoc apprenticeship" is a better term.
You're right. No company large enough to have an HR department is going to let some outsider come in and touch their network.
You've got to prove yourself. You're working in the business now. Find your company's Linux admin and offer to help. Make sure that he understands why you're helping. Tell him you're willing to help after hours -- maybe even off the clock. Start taking the unsexy jobs off his hands and work your way up from there.
And Apache is a good start, but you have a lot more to learn. You have to become a JOAT.
Learn everything you can cram into your brain, and understand the theory behind what you are learning. Networking and firewalling. Samba. NFS. NIS. RADIUS. Kerberos. Know installation and programming of MySQL or PostgreSQL, and be familiar with Oracle. Understand how to install and configure Apache. Understand configuration of at least one of wu-ftpd, proftpd, ncftpd and be prepared to support your decision of which you chose to learn. Know installation and at least minimal programming of PHP, perl, python, c-language, and three or four more languages of your choosing. Be able to compile software from source. Understand what daemons are safe to run on Linux and which aren't, and why, and how to provide similar, safer services. Installation and configuration of BIND. Installation and configuration of at least one MTA, preferably more. Installation and configuration of at least one of StarOffice, OpenOffice, the KDE office suite, and Gnome's office suite. Installation and configuration of at least one window manager.
Know the protocols behind what makes the internet work. Be able to send and recieve email and hit a website using nothing but telnet -- it's the only way you will know what errors are really cropping up.
And then if that weren't enough, learn Microsoft, too. Exchange Server, SQL Server, IIS, win32 OSes. It's still rare that companies are purely Linux.
Administration isn't like programming. You can get hired right out of college to program. The damage you can do can be contained as you earn your grey hair, and your damage can usually be undone. But if you do something dumb with a company's network, they're losing money. Immediately. So companies tend to be cautious.
Does any of this help?
Perfection in engineering does not happen when there is nothing more to add. Rather it happens when there is nothing more to take away.