The current record is, I think, about 0.3 AU or 45 million kilometres, so not exactly that close. There are plans to get a probe to within about 6 million kilometres (Solar Probe Plus, due to launch at some point next year apparently). Even at that distance conditions are pretty awful. I suspect that it will be a while before we can get closer still -- in part, at least, because most solar observations are made better at a fair distance anyway, so there's little incentive to solve the technical problems that are required to get, say, within the Corona.
All the same, given enough time, it might be plausible to do Star-Trek like runs to within tens of km of the surface. For the time being, the closest distance we can access is measured in millions of miles/ kilometres.