Naval Reserves
 

Is it possible to fly in the Naval Reserves off the street, similar to how getting and Air National Guard pilot slot is?

No.
Not like the ANG. Would have to come from active duty to fly in the Navy Reserves as a selected reservist (Part time like many in traditional Reserve/Guard squadron service member) . Know an Army helicopter pilot who was picked up after flying airliners to fly Navy Reserve C-9/C-40 (737).
Active duty Marine Aviators occasionally transfer to Navy Reserve after complete active obligations.
Good journeys

What Salty said. It used to be possible, decades ago, but now any flight training incurs a regular active-duty obligation.

I also know a guy who started in the guard, and then transferred to the same airframe in the Navy Reserve.

Generally its seems once you get mil wings, it's at least theoretically possible to switch between reserve branches (depending on their needs). Be aware that army helo pilots who did helo-only training (no T-34/T-6) would might need to attend UPT to get FW rated before joining another service.