You might indeed have suffered from radishes gone bad. When pickling or fermenting, don't eat something you don't trust.
However, it's also possible you simply experienced the wholesome stench of radish fermentation. From what I've read, fermented radishes are pretty well known for their rank smell. Some have compared it to farts or old gym socks; my wife says it's like decomposing cabbage. I can't disagree. Whatever it is, it's sulfurous, and radishes are a source of sulfur. I've pickled with fermentation, but the same probably applies for vinegar pickling.
However, I'm fermenting my second batch, and despite the stench both batches have been really, really tasty. My son likes them too, but I don't dare open the jar if my wife or daughter will be in the house sometime soon.
As for reducing the smell: I've seen some claims that peeling can help, but that didn't help me. I haven't found any other useful suggestions, and I suspect it's just the nature of the beast. My suggestions:
- To reduce the risk of actually-rotting veggies, follow a fermentation recipe. There are lots out there, even if not specifically for radishes. The basics: clean everything well, keep the veggies submerged during fermentation, and use a reasonably salty brine; I use a 3% salt solution but 4-5% is even safer.
- If possible ferment in a garage, basement, or somewhere else out of the way. The radishes will vent CO2 during fermentation, and wherever they exhale you can expect the noisome aroma. Only put them in the fridge when you're happy with the way they taste, at which point you can keep the jar tightly sealed.
- To share them: open the jar quickly (and preferably outside), remove a few slices, then rinse them well. Rinsing greatly reduces the smell without diminishing the flavor.
If you decide it's worth the trouble and the eye-watering funk... bon apetit!