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
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.
The issue is fundamentally that of "categorization": people want to have a box to put you in. "Dr. X is an expert in field Y." Early on, if you're all over the map, people don't have a clear sense of what your focus really is. That makes it harder for them to feel that you're going to be focused on their needs in your next position. Instead, the worry is that you'll continue to be all over the map.
This is also a problem for young faculty: they need to have a broad enough profile that they aren't trapped in a particular "niche," but not so broad a profile that they don't have depth in any one specific field. If someone can't be recognized as "the expert in her field," where 'her field' is somewhat arbitrary in scope, that makes for problems when it comes time for promotions and tenure cases.
It's true in an abstract sense that doing more is better than doing less, but there are psychological factors at play here.
Regarding extracurricular activities, hiring committees are unlikely to value them much, and they will come across as a distraction from research. For each activity mentioned on your CV or website, someone may read it and wonder whether you might have written another paper if you hadn't been doing this instead. It's not really fair, but you don't want people to be thinking about this.
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?
Partly it depends on how good it is. If you add a truly excellent paper to your CV, it should only help. However, research outside of your specialty or done on the side is probably less likely to be excellent, and someone who looks at just that paper may end up with a lower opinion of you than you would like.
I'll echo what the other answer have suggested and add a little more. On the academic job market you want to be able to explain what you do in a way that people can understand in a sentence or two. Your question seems to imply that already understand that having a focus is important and excelling in it is of utmost importance.
There are two ways that work or research outside of this core/focus can hurt:
Peripheral work may leave you with less time to make the core/focus really shine. You may simply have less achievements or publications than you would have if you had focused more on your core research. The issue is not only that people reading your CV might think this. It might really be true!
The second issue is that this peripheral work might be seen as a signal that you are not serious about your core body of research. Do you really care about devoting your life to the field, topic, or question that you are asking someone to hire you to work on? Are you likely to leave your career for this other thing? The core of your work might be seen as less focused than it actually is if it looks like you've got all these others things going on.
This second issue is a real risk, but it's possible to deal with this. Basically, it's your job to convey to people that although your extracurricular work is there — and although it may even constitute some impressive achievements or skills — you don't treat this other work as seriously as you treat your research.
This often means leaving irrelevant stuff off of your CV and website — although there are limits to what you can leave out. It also means organizing your CV so it's clear that the central thrust of your research is your priority. Many people have "selected papers" on your website or other personal materials. You can get to make that selection.
For example, I have written several technical books, served on several non-profits, and given hundreds of talks at (non-academic) technical conferences. I mention these things in brief and in passing at the end of my CV and on other pages on my website reserved for my non-academic work. I don't hide these achievements as I think they speaks to my skills and qualities as a researcher. But I make sure that when speaking to academic audiences, I — quite literally — place the core of my academic work first.