ESXi can be used for personal use at home for free.
You can install ESXi to any disk you like including USB drives and even SD cards, however your computer's BIOS must be able to be configured to boot from these external devices. Assuming your BIOS allows this use the ESXi installer ISO/DVD to install ESXi to your external disk. Then after the install is complete you can set your machine to boot from that drive and ESXi should fire up. Please check VMware's HCL to ensure your hardware is supported.
Assuming that all goes well you will assign an IP address to the ESXi host with an IP that your home network resides on, perhaps 192.168.1.X (example)
Once ESXi is running you can create a virtual switch that is connected to your computer's network card thus granting your virtual machine's access to your network. Assign your virtual machines an IP on the same subnet as the rest of your machines and you will be able to access them from your home network.
It should be noted that loading ESXi on desktop hardware is probably not the best idea. You would be better off with VMware workstation in that scenario.