how do you use time machine to backup multiple computers on a network WITHOUT a Time Capsule?

There are countless numbers of Network Attached Storage (NAS) devices, of which the Time Capsule is one. All of them connect to a network router using an Ethernet connection.

The important item for you to consider would be support of Time Machine. Not all NAS devices do. Synology.....a company that offers highly regarded NAS products....does provide full support for Time Machine.

To add to Bob's comments and from personal experience, I have been using Synology NAS devices for a number of years. I currently have a DS916+ and use it for Time Machine (TM) for the last year or so without any issues.

Since the release of macOS Sierra, I stopped using my Time Capsules (TC) for TM as they have become unreliable. Instead I continue to use the TCs with Carbon Copy Cloner as an additional backup solution and for file server services.

When Apple discontinues the Time Capsule, that does not mean that Time Capsules will stop working.

If your Time Capsule is getting old, you might consider buying a new Time Capsule now.....while stock lasts.....A new Time Capsule should go for another 4-5 years or more. The products appear to be selling briskly, and there are no discounts available from Apple that I can find.

Apple supports a product for 7 years after it is discontinued.

If you don't want to stay with Apple, a Network Attached Storage device (NAS) can be used to store backups of multiple Macs. The Synology brand in particular is recommended, since it does fully support Time Machine backups.

More info here:

NAS Selector | Synology Inc.

We are not able to recommend specific models, but this article should help get you started:

The Best NAS for Most Home Users: Reviews by Wirecutter | A New York Times Company

Click on the links for more information including a copy of the product user guide.