A doctor wants me to put a links page on his website to other medical institutions. I'm wondering link pages are even used anymore.
I use it on my personal website. It's just a collection of random links I find interestnig. Or perhaps soemone wants to backlink with me. I'll add it there. It's mainly just stuff I find relevant to my personal site (which is my hobbies), which in turn, are interesting to my users.
Sure they are. As said, make sure the links are relevant and useful, though. It's worth giving some information about how clicking the link will benefit the user, too. It can be part of a doctor's service to provide useful information for patients—but your client doesn't want people to leave his/her site for no good reason. All kinds of sites link to external sites to provide useful info to clients and customers. (This is supposed to be one of the main ways search engines like Google assess the usefulness and value of sites.)