I just use a mini pc I bought from AliExpress. It’s a GMKtec with an n100 it has SSD storage and plenty of ram. I use a Sonnoff ZBDongle-E for the Zigbee radio also from AliExpress. You can even flash the dongle to run OpenThread so you can use it for Matter. Though you need to buy two dongles to run both Zigbee and Thread. It cost almost the same as an HA Green.

It’s a protocol that runs over WiFi, Ethernet and Thread. So not Zigbee. Though the Zigbee alliance was part of the group that designed the Matter spec. Many Zigbee hubs also support Matter but the two protocols are on two distinct networks. And if you want to use Matter over Thread you need a device that can act as a Thread Border Router. Like an AppleTV or a Home Assistant instance with a Thread antenna.

or Matter

Yep I second this. If have a GMKtec NUC with HA on it and it runs perfectly.