How to become a DBA...the non-standard route (not beginning as sysadmin or dev)?
You can add a lot of value to your career at little or no cost except for some of your  time.  You need a desktop computer or server with at least 4 GB of ram. Download Oracle virtual box or pick a pre configured image here.

Install and configure.I learned more by breaking things than by reading in books or videos so don't be afraid to break your database repeatedly as you figure how to use the tools of the trade. With virtual machines you can take a snapshot, break the image and revert to the saved snapshot.

Once you have installed the database you can try adding a schema of your design or play with Oracle APEX to get a quick idea about web applications. 

All the Fusion applications are available for download like BEA weblogic and you can play with some Hello world apps in Java.

Then try and duplicate some of the things you need to do at work like backup, tuning, RMAN and anything else you need to practice on.  It won't be easy but it will reward you with hands on experience which will give you confidence for work tasks.