Looking back at this question, it occurs to me that going to the post office doesn’t necessarily involve spending money. If you have a post office box, you might just be going to pick up the mail. Of course, you’re probably paying for the box, but that isn’t a direct result of the errand. And you also might be picking up a package left for general delivery, or one that was too big to fit in your (free) mail box. The post office used to leave me little notes in my mail box when I got packages, and I’ve have to drive down and pick them up. Assuming the postage was paid by the sender, the errand didn’t involve spending anything but gas money. People used to go to the post office to pick up tax forms; they don’t do that now, but it is an example of a “cash free” errand.
Sometimes, we drop the kids off for a play date or friend’s birthday party, and we pick them up later. No cash outlay there.
Other times, my wife takes a meal to a sick friend. We pay for the food of course, but the errand itself doesn’t involve spending any money. Some other kinds of errand involving helping other people could be considered errands without cost. My mom has driven my grandmothers around to doctor’s appointments and hair appointments and stuff like that for years. They are spending money, but she isn’t. Her errand is to help them.
There are some things you have to do that are paid for by others, or that involve only filling out paperwork. When my dad became paralyzed, our local transit wanted to see him in person before issuing a special transit certificate. When I open a bank account, I’m tranferring money (into the new account), but not necessarily spending money—I have to go down to the bank and sign a signature card and maybe some other paperwork to get that done. Before the internet, there were a lot more paperwork-oriented errands, like filling out credit card applications, that would be done in an errand-like fashion.
When I was a kid, we used to grow a huge vegetable garden. We always grew more than we needed, and we’d put beans and stuff in paper bags and take them around to the neighbors. Cashless errand, right there.
Final thought; registering for your wedding gifts. Wander through the store, scan items with a bar code reader, spend nothing. Great errand.