Can I set up the HomePod mini when I am on vacation or something and not at my exact home?

Hi withers28,

Welcome to Apple Support Communities! It's a great question. You can certainly bring your HomePod mini with you and connect it wherever you have access to an electrical outlet and Wi-Fi. Your HomePod mini automatically connects to the same Wi-Fi network as your iOS or iPadOS device. To reset the HomePod and set it up again, you can follow the steps in this guide:

Reset HomePod - Apple Support

Let us know if you've got any questions.

Take care!

Note however this caveat from If you can't set up HomePod - Apple Support:

• HomePod doesn't support public or subscription networks with sign-in requirements. These networks are often in places such as businesses, schools, dorms, blocks of flats, hotels and shops. 

Guest Wi-Fi networks that allow a connection to the Internet but block peer-to-peer communication are also unsuitable for setting up HomePod.

tt2