I'm using Proxmox as my hypervisor, based on KVM & Debian. While less popular in enterprise, it's fairly popular for home labs and for SMB-grade infrastructure it works just fine. Additionally, it's based around open-source software and doesn't require licensing fees.
Network segments are created using Open vSwitch. The host has an IP on the LAN & SRV segments for now, but will be moved to just the SRV (proxied) segment soon. There are also DMZ & LAB segments that can be accessed one-way from the LAN but are otherwise isolated. Additionally, the LAN & WAN segments each have a physical NIC attached to them to allow communication with other devices, including the internet. The host is not directly connected to the internet.