score:2
The California State Parks webpage of the Old Town San Diego State Historic Park (SHP) says:
Park Hours
Visitor Center & Museums 10:00 AM - 5:00 PM
Park Restrooms Open 10am-Dusk; Concession Restrooms may be open longer.
Museums are closed Thanksgiving, Christmas, and New Years Day.
Thus, the governmental part (visitor center, museums) of the SHP closes at 5:00 p.m.
The SHP, however, is a built-up area of historic buildings and streets. This non-governmental page contains this illustration:
The illustration doesn't show pedestrian or vehicle barriers.
The earlier-cited SHP page says concession restrooms may stay open later. Restaurants and specialty shops and hotels are located within the SHP, and listed on this non-governmental page. I looked at a few restaurant listings, which show closing times of 8:00 and 9:00 p.m. The Cosmopolitan Hotel is shown as "open," while the Cosmopolitan Hotel Restaurant is "closed."
Looks like the Cosmopolitan Hotel is at this moment the only option for staying overnight within the SHP.