To find out if you should be quarantined you should contact your local health department.
Note that there are specific guidelines for employees who are delivering essential services - and this includes veterinarians who are working on those veterinary services that are exempt from Executive Order 202.6 (That closes non-exempt business premises)
Here are the NY State
Department of Health Protocols for Essential Personnel to Return to Work Following COVID-19 Exposure or Infection
New York City
Department of Health has issued it's own guidelines specifically for veterinarians.
We talked with NY DoH about this and they asked us to stress that contacts of cases should remain home if essential functions can be maintained without the exposed staff member being at the work place. Exposed contacts should not be at work unless it completely jeopardizes the ability of the practice to function. Clinics should be smart about who they allow to come back and when. The ability to have exposed but healthy contacts remain at home may differ in different parts of the state as the virus spreads across the state. Contact your local health department to discuss the epidemiology in your region.
The local health department will be making their decisions based on guidance from the NY Department of Health and you can download their guidelines.
Those guidelines require mandatory quarantine
for a "Person has been in close contact (6 ft.) with someone who is positive, but is not displaying symptoms for COVID-19"
That means that if anyone tests positive at your hospital, everyone who was within 6 feet of them during the last 2 weeks could be quarantined by the Health Department.
They also have a precautionary quarantine
for anyone who is known to have had a proximate exposure to a positive person but has not had direct contact with a positive person and is not displaying symptoms.
However another set of guidelines
to local health departments clarifies that for coworkers
If an asymptomatic employee calls their supervisor and notifies them that they are required to stay home in quarantine because they were a contact of a confirmed case, the contacts they had at work are considered “contacts of a contact” and therefore not at risk.
These “contacts of a contact” are not required to be in quarantine
and should be permitted to continue to work, following recommendations for social distancing as appropriate.
So operating safely isn't just about keeping individual staff members safe. It determines whether your hospital is going to be able to be around to provide essential veterinary care. If you can't provide that care, then there will be a greater burden on the hospitals around you that may already be stretched to capacity.