In a comment, Xombi asked about research you care about vs. putting food on the table. GMP recently also had a post on this topic. As GMP said in her comments, Dr. Sneetch pretty much summed up this issue in a comment--it is a lucky few that can spend most of their time on following their own interests. Everyone else has to spend most of their time on what funding agencies think is important.
Most of the time, I can find interesting aspects to fundable research that I enjoy. This is what keeps the job fun. One thing to keep in mind is that startup funds can seem like freedom to follow your research bliss. I think this is a big mistake. You need to use your startup funds to get evidence that your lab works on RELEVANT and interesting research. The data you generate is the preliminary data for your proposals, and has to show that you are fundable! My strategy was to select target funding agencies/program officers, and shape my starting research to show what I can do in their areas of interest. I picked areas that I am interested in, of course, but fundability has to be a bigger concern than personal interest if I want to keep my lab going.
In a National Lab setting, I found that I could carve out some time to work on my own research interests, especially before I became a PI. As a postdoc, you should have minimal responsibilities in supervision and proposal writing (although if you want to get hired, you need to be somewhat involved in this so you can see how things work). If you work efficiently, you can get your own work done, and then work on side projects. No one will really care if you use National Lab resources as long as it leads to papers/patents/funding. My side projects as a postdoc became the core of proposals (both mine and my PIs--you do need to show your sponsor that you add value after all!).
I don't think that fundable work is in opposition to good work. It is possible to do good, important work that is of interest to funding agencies, and I think this is what most researchers try to do. There is some room to shape the scope of the work after the project is funded, but of course the goals of the program officers need to be kept in mind if you ever want to be renewed/funded by the same agency again! But this shouldn't impact the quality of the science in the end. Just the excitement of the PI.