Ironically without an appropriate degree it can be harder to work your way into an average government contractor than a big tech company. In particular, I could see agencies like NASA strictly requiring an accredited degree as I know a lot several government contractors to do. I have a desire to someday work in space exploration, and have to make sure of knowing JPL & NASA's requirements ahead of time.
I have a Arts degree. I spoke with someone with prior experience at NASA. From what he told me, I would say their SWEs have backgrounds "diverse, but still very same". They come from many degrees, but they're all STEM degrees.
Assuming they don't have strict degree requirements, I could lay out a job path that goes from local web shop (I'm mostly a web developer) to local web & app shop. At this point I get to add Java to my resume for Android. Even though Android is going away from Java it will be years before it stops being used.
After 1-2 years of java experience I'll likely be more applicable for a lot of good software engineering jobs and on fairly equal footing with someone with a 4-year degree. I might still have that barrier of entry with JPL, but I'll cross that bridge when I get there.
Now let's go to the other option, getting a more appropriate degree. I would like a second 4 year degree, perhaps in ECE or CS to complement my Art degree.
A lot of people also want to get another degree, but that's easier said than done since some options become harder and some become easier when you are in your mid 30's like me
I have no SO and no kids, so potentially I can be very mobile, but the lack of a job offsets that. How do you set aside years of time for another full degree, with a small support network, and prevent myself from living on the streets? I don't have any friends or family that I know of that'll let me life rent-free for years. I also have to consider if I have to return to the life of working a humbling service job. I'm not in my early 20's anymore so I can't be as flexible and open with part-time jobs and living arrangements.