Radiometric dating data sets
It sounds like there are two primary ones:


No, these are not the two "primary ones". The method used depends on what you are dating, and what age you expect it to be.

Radiocarbon dating is relevant to things younger than a few tens of thousands of years, and it's only relevant for things that were living (or growing), and incorporated atmospheric carbon.

Other methods such as U–Pb (the "gold standard" when dating igneous rocks), Rb–Sr, Re–Os, Sm–Nd, K(Ar)–Ar, Lu–Hf, and many more, are used for older things in the millions to billions of years. The one used depends on what you're analysing (Re–Os for sulfide minerals, U–Pb for zircons, Rb–Sr for micas, Sm–-Nd or Lu–Hf for garnets), and what is present in the rock, and what has textural evidence for being preserved.

You might call U–Pb on zircon the "primary" one, because most of the dating is done on zircon. It's robust, easy, gives good ages, quick and relatively cheap.


  First, I am wondering if there are any data formats used for storing the data (which will help with search). 


Most U–Pb data is reduced using IsoPlot, so the data will look like something that came out of that software. But in general, no. There is no standardised  data format for geochronology, unfortunately.


  And third, if there's not any "centralized repositories" of this sort of data


There are several of those. You can search EarthChem, that has some geochronological data. Australian government agencies also have some geochronology data in them, for example GA or NTGS.


  If it matters, I am interested in particular in fossil radiometric data.


Fossils are among the hardest thing to date. If the fossils are relatively young (few thousands of years), there's radiocarbon. Anything older than that, and it becomes much harder. There's U–Th disequilibrium series which may work for some slightly older fossils (~1 million years). Anything older than that in deep geological time has to be dated with indirect methods. Finding a database of that is probably wishful thinking, and to understand their dates you need to have some geological background to understand the geological relations and considerations that allow the indirect dating.