The interview process was extremely pleasant. I submitted my resume to their website and got a call back from a recruiter within 2 days. The recruiter set up a phone screen with a senior engineer 2 days later. The phone screen lasted about 20 minutes and the interviewer told me that they wanted an in-person interview before the phone call was over.
My in-person interview was a couple of weeks later. SpaceX put me up in a nice hotel in Manhattan Beach, a few miles from their headquarters. The interview started off with a tour of the factory, where you get to see the rockets and dragon capsules being assembled. The tour was incredible and worth spending a day interviewing on its own.
The first step of the real interviewing was a short electronics knowledge test, think about calculating the gain of an op amp and things like that. After that there was some practical knowledge demonstration, such as inspecting some solder joints and identifying the flaws, finding a signal on a spectrum analyzer, and identifying potential electronic noise sources. Beyond that there were several interviews with members of the team. These segments mostly had 2 engineers asking questions back and forth. The questions were fairly in-depth, but if you know your stuff then it's not bad at all.
Overall it was a full day of interviewing from about 8:45-4:30 and about 10-11 people total. Everyone was very pleasant and it was obvious that all of them were very smart and know their field. I walked away from the interview feeling like I just spent a full day hanging out with colleagues. They did a great job of not making me feel nervous or overwhelmed. The whole interview process made me want the job even more than when I initially applied.
After the interview be patient. It took right at a month for an offer to come through. My advice would be not to think about it too much after your interview. I was slowly driving myself crazy waiting for the phone to ring.