If you can't meet your friend for dinner on Tuesday because you have other plans, you are unavailable. If the shoes you want have been sold out, they are unavailable. Unavailable means not accessible.
If you avail yourself of something, you use it or take advantage of it. You might avail yourself of discount tickets to the theater provided by your school. If the tickets are all sold out, they are unavailable to you. In the context of dating, if someone is unavailable, he or she is already in a steady relationship or married. Someone emotionally unavailable is not open to sharing their feelings.