If you have multiple devices on your local network, then you must have a router between your ISP's connection and those devices supporting NAT. Generally these routers use a private subnet like 192.168.1.0/24, with the router itself using 192.168.1.1. But you'll have to check your setup to confirm.
Either way, assuming that is correct, your router will have DHCP settings, and you can set it's starting address to a higher range so you have some room for static IPs. Otherwise you may run into IP address conflicts on your network.
What static IP are your trying to set on your ESXi host? I would speculate it's changing on your because what you entered was not valid. Either way, should not be a need to get static IPs from your ISP.