Are there objective ways to measure narcissism?
The current diagnostic criteria of narcissistic personality disorder (NPD) requires five out of nine indicated signs (citing DSM-5; APA, 2013):

>1. Has a grandiose sense of self-importance
2. Is preoccupied with fantasies of unlimited success, power, brilliance, beauty, or ideal love
3. Believes s/he is special and unique and can only be understood/associate with other special or high status people/institutions
4. Requires excessive admiration
5. Has a sense of entitlement
6. interpersonally exploitative
7. Lacks empathy: is unwilling to recognize or identify with the feelings/needs of others
8. Is often envious of others and believes others are envious of him/her
9. Shows arrogant, haughty behavior/attitudes

To objectively measure narcissism, there are psychometric tests such as the Narcissistic Personality Scale for this matter. However, research advances challenge the lack of empathy in narcissistic people: stating that an identifying feature of NPD (lack of empathy) is probably misuderstood. So in essence, the tipping point that separates what is normal from what is narcissistic is being shaken again.

Is it when it becomes a problem for others? For ourselves?

This is generally correct, depending on the perspective. A clinically diagnosable disorder is generally defined as a condition that "causes clinically significant impairment in social, occupational, or other important areas of current functioning.(APA, 2013)" You can see that diagnostic criteria in most mental health disorders like mood disorders (depression), anxiety disorders, trauma-related disorders, etc.

However, this criteria isn't present in personality disorders. The unifying feature of personality disorders is that it has specific pervasive patterns of behavior that begins by early adulthood. That's it. I am convinced to think that your question is still under debate of researchers worldwide.

However, in practice, clients seek the diagnosis for personality disorders when it causes significant distress on them or on other people. So it becomes NPD (or other PD) when it becomes problematic.