Visit the Hermitage
It is entirely possible that you are already hiding from everyone today. If so, keep up the good work.
If you're not yet in hiding, run. Run away. Retreat.
Attending a retreat will not serve the purpose of this day because every retreat, almost by definition, involves meeting with other people who want to "get away" for a similar purpose.
Hide from Everyone Day is a day for solitude and reflection. Go off to a place with a mirror or a lake and look at yourself.
You can do other things today too, as long as you do them by yourself.
This is one of the primary reasons the Internet was invented. You can get pretty much everything you need (especially if it's just for a day) by using your Internet-connected device.
If you order out for food, you might have to meet briefly with the delivery person, but I guess that probably acceptable.
Most likely you have enough food already in the house to last for the day, so that exception to the rule might not even be needed.
Enjoy your day of talking to yourself. Just don't make it a habit.