Recently, I went to a public aquarium. Another guest at the aquarium mentioned that she wanted to get a small shark as a pet. I know most people are not prepared to care for larger sharks, such as blacktip reef sharks or leopard sharks, but would a small shark make a good pet?

No.  Sharks generally make poor additions to home aquaria.

The wikipedia page on the matter is pretty good:

http://en.wikipedia.org/wiki/Sharks_in_captivity
 

Note that the small freshwater 'sharks' commonly sold in pet shops are not actually sharks, but usually a form of Minnow.

>Note that the small freshwater 'sharks' commonly sold in pet shops are not actually sharks, but usually a form of Minnow.


...and just to add that although small compared to real sharks, some of these freshwater fish like the Bala shark Balantiocheilos melanopterus, and red-tailed shark Epalzeorhynchos bicolor actually grow rather large for tropical fish and are probably not ideal for inexperienced fish keepers. They can and do make good pets provided you have a big enough tank and the knowledge of how to look after them.

To summarise my answer regarding sharks as pets: "No no no no no no no no no no no no nooooooooooooooooooooooo!!!!!"