I worked for the DoD for 10 years, as well. It was my first job out of engineering school. I had a Civil Engineering degree, and I was hired as an environmental engineer at one of the Air Force's large Superfund sites. This was in 1990, when Regan was out and Dubya was in. Of course with the politics at that time, big government was bad, bad, bad, so there was a hiring freeze on. That posed a small problem for the department that wanted to hire me. See, they were under this hiring freeze, but they needed someone to do the work, or face fines of $10,000+ per day from the EPA and state for not cleaning up the mess they made of the water supply. So what can they do: hire contractors!
I worked for the University of Tennessee, Knoxville as a "graduate intern". (To this day, I've never been to Knoxville, TN, by the way. The job was in California.) My first day on the job, I was tasked to walk paperwork around the base to get about a dozen signatures. It was the paperwork that secured the funding for my position. Our salary was about $27,000 per year at the time. The paperwork in my hand said they were paying about $65,000 per year for me. It was a similar situation to the OP - the government was providing for all computers, offices, and other overhead. Ooooo - bad contractor taking the US taxpayer to the cleaners, right? Not necessarily...
See, to do any procurement - i.e. let a contract - is a major hassle, involving lots of regulations and procedures. It would take in the neighborhood of 2 years to go through the process from start to finish, and take lots of government employee man-hours (they really did have other real hazardous waste cleanup work they would prefer to be doing). So it is common practice in these situations to look around in the government and see who _already_ has a contract, and piggy-back on that. Well, as it tuns out, the Department of Energy had a big contract with Martin-Marietta to run Oak Ridge National Labs, and Martin-Marietta has the University of Tennessee, Knoxville (UTK) on subcontract. So, this Air Force base in California had a Memorandum of Understanding with DoE to use their contract to get warm bodies through UTK.
So who's getting all of that extra money I'm not seeing? Well, each layer in the contracting process takes it's own service fee for managing this arrangement. Marint-Marietta adds their percentage, the DoE adds their percentage, DoD adds their percentage, HQ Air Force adds their percentage, our command (AFMC) takes their percentage, and my base adds their percentage. After all that, we're up to $60,000+ per year for a graduate intern, so the base can avoid $10,000 per day in fines and do the cleanup work they should be doing in the first place... because some bonehead politician has to cater to a constituency that whines about big government, implements a hiring freeze, and still demands that that same government fly big, expensive (but very technologically cool) machines around to blow up people.
Now, the OP was likely in a somewhat different position, but given my experience, it's not surprising. It's been going on for a long time, and there are reasons for it. Not good reasons, but insting that the problem is simply "big government" and "greedy contractors" without looking at our own expectations of that government is stupid.
Incidentally, a year later I was hired into the position as a Federal employee, and took a $500/year pay cut. I wrote the position description for the job I was applying for, and spent about 4 month shepherding that paperwork through the process, so the department could get someone who they know could do the job - me.