Funny story (BTW, LBNL isn't a "nuclear weapons lab" -- they moved all the classified work out to Livermore and Sandia years ago) about that, though yes of course the UC has the contract for administration of the labs. Berkeley is indeed, by city ordinance, "nuclear free" -- but there are some exclusion zones due to the fact that local ordinance can't override state and federal law: LBNL; UC Berkeley itself (where there's plenty of nuclear research going on; they did shut down the reasearch-grade reactor a while back, but because it was old, not because of any pressure from activists or the City); and the portion of I-80 that runs through the western edge.
In other words: all the places where you'd expect there to be nuclear materials.
But: if they catch you with some cesium in your pocket while you're walking down Telegraph: bam, you're gonna get a ticket! $500, baby!
/jordan
http://www.ci.berkeley.ca.us/bmc/berkeley_municipal_code/title_12/90/index.html