There are some who feel everything. They may be overwhelmed in a crowd. There are others who feel "nothing". Most of us are somewhere in between.
Empathy is the ability to identify with or understand another's situation, feelings and motives. This may be past, present or future. The attribution of one's own feelings to another person, animal, plant or to an object may also be called empathy.
Many who are empathic pick up the feelings of others intentionally or not. Medical practioners, for example, tune into the feelings of those they are helping. Some shut this ability down because they don’t know what to do with it. Others are continuously overwhelmed because they don't know how to handle it.
And somehow, some who are highly sensitive manage to block out the distressing feelings of others. But these are few. However we can all learn to minimize our own emotional reactions.
See: Boundaries for more information.