I used to live in a house with 1960's plumbing. All the faucets (except the kitchen sink) had washers. I eventually became expert at replacing washers.
Does your leaky faucet valve have washers? If so, remove the valve core (the part the washer is attached to) and take that, with the washer attached, to the plumbing supply store. I used to think all washers the same diameter could be used interchangeably. No, different shaped washers are needed, depending on the make & model of the faucet. The guys at the shop, if they're any good, should be able to look at the core and tell what type of washer you need.
They can also look at the washer and see if you have to replace the valve seat. That's the removable brass ring the that the washer pushes against when the faucet is off. These seats can get nicked or worn so even a new washer leaks.
In the 80's I replaced a number of those seats and then the faucets were as good as new. Still, most hot water faucets needed new washers every 6 months, and the cold water faucet washers lasted about 2 years. When I moved to my present house in 1993, all the faucets were washerless, an I lost all my faucet kung-fu skills.
