Karen Lewellen wrote: 
> I believe I should have been more specific.
> Yes, I  own domains.
> What I do not own is an IP address.
> It is the process of securing one I am asking about.
> I am not sure the provider I am going to change to does this any longer.


Inside a home (or business), it is normal to use the private IP
addresses reserved for that purpose. The router or firewall
connecting to an ISP will translate them to a public address
supplied by your ISP through a process called NAT: Network
Address Translation.

https://en.wikipedia.org/wiki/Private_network

When you only have one network-attached device, no translation
is needed.

The most often used private address ranges are:

192.168.0.0/16

and

10.0.0.0/8

There are dynamic methods of requesting an address (DHCP) or you
can choose to permanently assign addresses. For a very small
network like yours, permanent (static) addresses are a good
idea.

10.0.0.1 router
10.0.0.2 DOSbox
10.0.0.3 Linuxbox

The two major methods of assigning human readable names to IP
addresses are the Domain Name System (DNS) and a hosts file.

For three boxes, using a hosts file is easiest. The downside is
that you need to write the same values on each computer, and if
you change them, you will want to reflect those changes on all
boxes. For a three box network, this is not time-consuming.

Can you identify the networking software you are using on the
DOS machine? It almost certainly uses the same hosts file format
as Linux, but will put it in some idiosyncratic place in your
filesystem.


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