I took a postdoc at a National Lab (NL) after completing my PhD. I was really tired of academia, and of universities in general, and I thought I wanted to start a career in industrial research. I mostly interviewed for industrial positions--the interview at NL was mostly a "why not?" interview, and not a planned move. However, the research was a fascinating new direction away form the sub-field I worked in as a PhD, and taking the position wouldn't preclude going back to industry when done, so I took the position at NL when it was offered. I was really excited to move my research in a new direction, and the NL postdoc was a great choice for me.
There are a few advantages to doing a postdoc at a NL. First, the pay is excellent. Like $60k+ excellent with full benefits. This is more than some starting assistant profs get! That said, the main benefit is in the resources. At most (all?) NLs, it is much, much easier to buy equipment than to hire people. Budgets are annual, so hiring people means supporting them. Many NLs are soft money only, or soft money for everyone except staff, so hiring can be risky for a PI. Furthermore, there is overhead of 200-300% or more on salaries but no overhead on equipment or supplies. This also means that your supervisor will let you buy things that will save your valuable time, rather than trying to save money by preferring cheaper but labor intensive protocols.
An additional benefit is that most of your colleagues will be PhDs or experienced techs. Many are experts in their technique who have many years of bench experience. Projects have a very fast ramp-up time, and usually only last 2-3 years. Independence and self-motivation are greatly prized. Your colleagues will expect you to contribute your part from day 1. The best part is that even though as a postdoc, you are hired for some specific task, you can pretty much work on whatever you want as long as you are getting your assigned work done. Furthermore, everyone at the lab needs to get 2 papers a year, so people are inclined to help with projects in exchange for authorship. It is much more straightforward to establish a track record as an independent researcher under these conditions if you are talented, independent, and motivated.
Other minor advantages--you can attend lots of conferences. I went to 2 per year as a postdoc, plus grant agency review meetings. We got a decent amount of good seminar speakers (but not as many as at a university). It is fairly easy to get both industrial and academic collaborations going, since you are viewed as less competition for IP as a government employee (the Federal government has an automatic license for patents, so collaborators have right of first refusal to pursue patents). If you do get a patent from your work at a NL, the inventors split 15%, which is quite generous.
That said, there are also significant downsides. First of all, if you intend to go into academia, your postdoc advisor will be of limited help. S/he won't know how to apply, how to help you with your application materials, and may not know the style of an appropriate letter of reference. There is a lot of deadwood in a NL. These people do minimum work to get their 2 papers a year (which don't need to be first author papers) and act as a research drag. Getting them out of your way can be difficult. In addition, many staff scientists prefer a hands off approach to mentoring because they are used to working with peers, not students or mentees. If you are not independent or a self-starter, you can float aimlessly for a while. There are few dedicated techs for instruments, and no students (except for summers, and with some exceptions), so you need to run everything yourself or convince someone else to do it for you. You will be competing with academic groups that can throw much more manpower at a problem, so you have to hope that being smart with fewer, better trained personnel outweighs the brute force approach possible with an army of students.
Where do people go after a postdoc at a NL? From my cohort of postdocs, I'd say ~25% stay at a NL as staff (this varies by lab, but many labs like to try before they buy), ~25% go into industry, ~20% go into academia, 15% go into non-research science positions (working for journals, granting agencies, the patent office, or other government non-research positions), and 15% leave science all together.