Why the embassy is closed on a day that is not a holiday
It is a holiday somewhere the embassy answers to. Missions observe the public holidays of their home country and of the country they sit in, because their staff are entitled under one calendar and their visitors, guards, banks and couriers work to the other. The union of the two lists is long: a mission can close twenty or more days a year beyond weekends, and the closures rarely match the holiday list of the country you are standing in.
The weekday closures nobody expects
Some missions also close on a fixed weekday every week, or take submissions only on certain days; closings trackers list missions that shut every Friday, and others that accept documents Monday to Thursday only. Add half-days before major holidays and emergency closures, and the working month at a busy consulate can be several days shorter than the calendar suggests. Appointment systems quietly absorb this: the missing days simply never appear as slots, which is why a fully booked week sometimes means a closed week.
The check to run before you book anything
Three steps, in order. Find the mission's own holiday list for the year, usually on its website; our directory records office hours for 293 of its 1,457 missions and links onward for the rest, because the mission's page outranks any aggregator. Cross out both calendars' holidays around your dates, plus the mission's weekly closure day if it has one. Only then book travel, and keep a buffer day: a closure discovered at the door costs a trip, while a buffer day costs a coffee.
Closures are a system, not bad luck
Once you see the two calendars, the pattern is predictable a year ahead, which is the whole point of checking. The directory gets you the right mission; its holiday list gets you the right day.