Do extracurricular research/activities in graduate school negatively affect your future? I'm the kind of graduate student that finds many research topics interesting and wants to participate in lots of student organization activities related to science and academia. But recently, one of my professors warned me against "doing too much" beyond my research focus, both in terms of publications and in terms of extra-curricular activities.  As I see it, your goal as an academic is to develop a "specialty", so it is important to focus on one narrow topic and pass over opportunities to research other interesting, but unrelated topics.  But can research outside of your particular focus in graduate school really negatively affect your ability to get hired in an post-doc or tenure track position in the future?  How can "doing more" reflect negatively on one's self?
Read broadly, publish narrowly deeply.

This is roughly what I've been telling my students. Now all of this depends greatly from area to area, but here's what I believe to be true. Having a broad background in your area might slow you down initially when trying to publish. But over the long term (your entire career), a broad base will help you more - it will let you be flexible about topics of interest, it will allow you to see connections where others might not, and it will help you place your work in a larger context. 

But from your question, you appear to be referring not just to "exposure to outside topics" but "activities related to the larger enterprise of science and academia". With those activities also, you should be careful. Maybe choose one or two outside activities and devote your extracurricular efforts there. The advantage is that by focusing, you're more likely to be able to do something meaningful, and it also prevents you from frittering away time in busy work.