The reason this was easily possible in my case is because of the peculiarities of bioinformatics. Namely, equipment costs are low, so there is more money for staff per grant dollar.
We also have had much more luck with center and collaborative grants than R01-style individual grants. On collaborative grants, our sections are almost always the highest scored, on top of the fact that the pay lines are inherently higher. On individual grants, it is hit-or-miss to say the least, because reviewers seem to think, somewhat rightly, "why do you need $1M to play on computers?".
People come to us and say, "hey, I want to put you in as a small section for $75K, $100K, or whatever". Over the course of time it has meant we have a diverse and relatively stable revenue stream and low overhead, and there is less risk (because less time spent) on each individual grant. Although amusingly, my PI is still associate despite pulling in way more money overall than many fulls, because of this idea that only PI grants count for promotion.
> I say that because that means I now have to pay for both of us while you go do science.
Yeah, although I said in my OP that "I just want to do science", in reality I write parts of grants, mostly the technical parts, all the time, and I go to grant strategy meetings, where I feel my input is valued as I can often give a better idea of what's feasible than the PIs can. I guess more precisely, I don't want the responsibility and pressure of being in charge of the whole package, and I don't like dealing with all the paperwork. So in our case the funding is a team effort as well.
> way things like training grants are structured
I have been with him since grad school. In the early days, the way we got around this was titling me as a postdoc and paying me like a staff scientist, which was fine by me.