The area can still be accessed with vehicles that have permits and a remote that opens it. Those are mostly:
1.) Delivery vehicles early in the morning (~6am) and late at night (~1am+) can come through to supply businesses in the closed area. They have a remote that opens the street blockages.
2.) If an oversized personal delivery needs to be made to a resident, it's possible to get a onetime permit that allows a truck to access the area.
3.) For people who have trouble walking the city has provided a few free electric powered golf carts/taxis that you can call and they come pick you up and deliver you to a destination (picture: https://www.visitljubljana.com/assets/Ljubljana-in-regija/Ka...). Since they're small and quiet they don't bother pedestrian traffic.
I wonder if Uber/Lyft could provide such service in US - they're already doing electric scooters, so why not electric transport for car-free areas?
Note that this is strictly old town center which has relatively small amount of residents but it's mostly dominated by businesses, hotels and airbnbs these days. The area is small enough to walk across in ~15minutes by foot. Near the edges there are underground parking garages so you can reach it via car (since public transport in the city isn't that great).