Almost every part of ordinary sink faucets can be purchased as replacement parts today. Of course you can buy the washers, even the "seats" that the washers push against and then grinders to "dress" the seats. But now you can even buy the handles themselves, as well as new "Hot" and "Cold" inserts to the handles.
The photo shows a special "handle puller" that can be a great help in trying to get old stuck handles off the faucet.
Specialty plumbing stores have always had much of this, but now the plumbing section of the renovation centres are carrying a rather complete line of replacement parts and specialty tools.