You can use a food processor, or a mortar and pestle, or a bowl and wooden spoon. Some use vegetable oil rather than olive oil for a gentler result. I do not. Your mayonnaise should have that bitter olive taste. Some thin with water; I feel this should be avoided.
Place the egg yolks, mustard and a gesture of salt in the food processor, mortar or bowl. Mix, then drizzle oil in very slowly, especially at the beginning so as you achieve an emulsified mixture. You can add up to 2½ cups oil; if you feel confident that the emulsification will hold, continue adding oil. If it is getting too thick add a splash of lemon juice and white wine vinegar; you can always add more oil. Season to taste.
After a while you will learn the various noises mayonnaise makes in the making which tell you when you have enough oil; these are hard to describe in words so I’m afraid you just have to listen to it. You want a consistency that has a body to it, but a body with give, not one which goes boing when you put a spoon in it.
If you are using a mortar and pestle, or bowl and spoon, follow the same process, but I stress the importance of the most gentle of drizzling of oil with this method.