This is something I've been thinking about quite a bit and I've decided that it is feasible. I've decided that since Linux, the BSD's and Unix has still not widely been adopted for personal use, the userbase is pretty savvy, ie people do not run their webbrowser as root, nor does their user belong to the root group. Secondly, if you click a link, most of the time you will be prompted to save to disk or run. Not to mention that I'm not aware of any mime-type for a non-windows executable or shell script, which will always bring up the save to disk or run dialog.
If for some reason, however, a mischievous file were to be run as a regular user, there is not much you could do. I guess you could place a mischievous daemon in ~/bin and place a line to run it in ~/.bashrc. At this point there's not much it can do. It wouldn't be able to run any pop ups since it won't start when X starts. I guess you could run a scan of the home dir and send it out to the bad guys and possibly send out netstat data.
Of course, if someone were to be running as root and such a program were to be installed, it could then be installed more covertly in /usr/bin and launched from /etc/init.d. Then nothing is really safe. There are a few strikes against this though:
1) For it to totally be generic for any Unix style OS, the installer would have to be a script. A BSD executable will not run on HP-UX. Of course you could target Linux users specifically, however, you're still making certain assumptions about which libraries are installed on a target system.
2) At this point you still have to be root and allow the installer to run. If you do this, it's you're own damn fault.
As Linux becomes more widely adopted and gives the users more conveniences, this can become a problem. So as an answer to your question: in the current state of Linux, BSD and the Unices, no you cannot get spyware.