A good starting point for people coming into the industry is usually tech support (L1). Maybe you can find a position in a company where they use many FOSS applications (they are quite widespread). From there you can specialize yourself and transition into a system admin.
But keep in mind that "breaking the power of tech gigants" will most likely not pay well (unfortunately).
Edit: depending on your degree you might also fullfill the criteria for graduate training programs. They will usually put you in a higher ranking position in a shorter period of time.