In terms of day to day, the life of a postdoc at a National Lab (NL) is very similar to that of a postdoc in academia (as far as I can tell from observing postdocs when I was a student), except there are no students. This means that postdocs are at the bottom of the status chain, except for summer students.
Once you change over to staff, though, your day to day experience changes a lot. First of all, now you are required to do service commitments. This can be serving on committees, doing science-related administrative tasks (safety coordinator, internal review organizer, running the seminars, etc), and is kind of similar to academic service. This is supposed to take up 20% of your time, but as always, it varies by whatever the task is. As a new staff member, you are on probation. Your probationary period lasts 3 years, after which you are much harder to get rid of. It is kind of like tenure, but the bar is much lower. Just do your job, and you are pretty sure to make it.
As a staff member, you are now required to have at least 2 publications per year (at least one is supposed to be first author until you reach a managerial level). Furthermore, most staff members work on 2 or more projects. This is by design at my old lab, where we could only pull 50% of salary from any one grant. You also need to raise money. At many labs, you are soft money, and must raise salary + 200-300% to be fully covered. You also need to cover whatever supplies you need (though at the lab, you will likely have access to any equipment you can think of, so you likely won't need to travel unless you need something really specialized). This can be difficult in times of lean budgets (like the past decade in physical science). As a result, I knew people who were paid from 4 different grants. They were supposed to work one day a week on each project, plus one for service (which is just insane!). Once you raise your own salary, any additional money can be used to pay other staff to work on your project, or to hire postdocs. If you are at a hard money lab, you just need to raise money for postdocs and supplies.
You can pretty much work on anything you can get funded, which is great. However, the high overhead means that it is not worth the effort to apply for anything less that $300k/year or so. If you don't have enough funding to work on your own projects, you need to find something else that has funding to work on. In practice, this will be found for you, but it may be something you don't want to do. This is how deadwood gets traded around form project to project.
Where can we get money? Lots of places, but some won't pay federal salaries (even if you are soft money) like the NIH. Staff at NLs compete for lab directed funds (LDR or internal research), which cover 50% or so of salary at my old lab. The rest comes from DoD, DoE, industrial collaborations, DARPA, USDA, NASA, and other similar agencies that will fund NL research. NSF is strictly educational (as so off limits), and many Federal sources are not really large enough to pay anything other than a postdoc in practive (NASA, USDA, FDA). Fortunately DoD and DoE have some "intramural" funds which are designated for NL research. On the whole, the job is a lot like an academic position, only you will still work at the bench for quite a while after becoming a PI. This is what makes it great training for a future academic career!