What can I do if there's not enough space on computer to backup iPhone?

I'm trying to backup iPhone 6 Plus, to them get all my info on iPhone X. When I try to backup, it says there's not enough space on the computer to backup. It's not my computer so I can't delete anything. What are my options?

Get a USB memory stick that is large enough to hold the data you want to back up (I have a 64 GB stick myself, but I think they go at least as high as 120 GB) then back up onto the memory stick. Since it is not your computer, then you won't fill it up with your data and you can take the memory stick with you when you return your computer.

If the data is valuable, make two copies on two separate memory sticks.

You'd have to use another computer or use your iCloud storage, though that likely wouldn't help in this case for the same reason.