If you search for private vpn static ip, you'll find hosted VPN services you can log into from anywhere, that will give you a static IP. These are likely to be faster, more reliable, and less setup and maintenance hassle than rolling your own. It looks like these services go for just under US$25/month, which you should see if your employer will re-imburse you for (it's their "static IP" demand that would cause you to incur this cost, after all), or at least you could write it off from your taxes as a business expense.
If you want to roll your own, you could go at least two ways.
One way would be to make sure you have fast and reliable home broadband service with a static IP, and you could set up your own VPN server at home. Perhaps your home gateway already has this feature. Or perhaps you could get that feature by putting aftermarket third-party firmware (e.g. OpenWRT) on it. Or you could buy a different home gateway that can do it. Or fire up VPN server software on one of your machines at home.
Another way would be to turn on VPN service on a colocated server or server VM instance at some service provider (and pay for a static IP address there, too, if necessary). That would likely be faster and more reliable than the "at home" solution.
Once you consider the cost and hassle involved, it might be better to pay for a "personal VPN with static IP" service as I mentioned above. Make your employer pay for it, or tell them to go pound sand with their "static IP" request. :-)