How to save data from API to state in Reactjs?

This is not possible.

State is maintained for the lifetime of the application. When you refresh the page, you are destroying the current instance of the application and creating a new one, therefore state can not be persisted.

The only way to persist data across a refresh is to use third party software or things like browser storage.

Use localStorage API https://developer.mozilla.org/en-US/docs/Web/API/Window/localStorage

You can put your data in localStorage and get it from there if you need it. For example

          if (!localStorage.getItem('data')) {
              axios.get(`https://example.com/new/${numberID}/staff`)
                  dispatch({
                      type: "FETCHSUCCESS",
                      payload: response.data
                  });
                  localStorage.setItem('data', response.data);
              })
              .catch(error => {
                  dispatch({
                      type: "FETCHFAILURE"
                  });
              });
          }