I'm curious if the Radioactive Sponge positive quality is worth making a character with. It's a quality in the hazardous environment section of Run Faster, and it has two effects: First, radiation hazard levels are one less for the character, so they are never in a lethal environment due to radiation (they do get sick). Second, they absorb that environment and spit it back out again for a while after being in it.

Can a decent character be built with this idea? I imagine you could get a radiation shielded set of clothes to contain your radiation, and they take it off when you want to be hazardous, but I can't imagine it would be effective on a scale fast enough to aid runners outside of very specific situations where you want to freak out people with radiation detectors. Even then you are likely to get a dedicated anti-terrorist squad or worse involved if they think a dirty bomb or nuke is involved.

Outside of using the quality with NPCs for a nasty twist on a mission (extraction or capture of a radioactive target)... what good is it?

It seems like one of those niche qualities that really is better off reserved for NPCs. As a player, you can and do see your character in all sorts of environments and while it might be nice to have some level of immunity to the bad parts of a given environment, that always seems to come with a load of penalties in every other environment.
You listed some of the big ones already, like the fact that you are setting off rad-meters everywhere you go. This quality on a PC would be fine if you were only playing in the barrens and around places where radiation was a large factor. Otherwise it seems more of a hindrance than a boon as you have to deal with the down sides and would rarely get to use the bonuses.

Well, as written the extra duration for your personal radioactive contamination only applies in Deadly-rated areas. So if your character regularly spends time in radioactive area, but stays out of Deadly environments like everybody else, it's a clear win for just 5 Karma.

Sounds like something for runners in Los Angeles, no?

Sounds good for someone from SOX or the Glow in Seattle. Especially nice for a toxic mage NPC.