Is there a way to save/backup packages previously obtained using apt install from a now dead repository?

A while back I installed a TV server program called Tvheadend using a PPA. The installation instructions and the program were on the site tvheadend.org which now appears to be dead (as of about three days ago, it seems). One of my pet peeves about installing software using apt is that you don't get a compressed installation file that you can hang onto for future use, like you would in Windows or MacOS. But what I was wondering is, does Ubuntu actually store the packages it downloads somewhere on the system, in a format where you could take that package and save it to an external drive or network share and then at some later time install it on a different system running Ubuntu, or even on the same system after doing a Ubuntu major version upgrade? I'd really like to be able to back this program up so that if I ever want to build another server I can still install it, but if there's a way to do that I am just not aware of it.

If there is not a way then that is really bad. I wish that Ubuntu would let you download some kind of install package to disk and then run that to install, as is the norm in Windows and MacOS, ESPECIALLY when you have to use a PPA to get software from a non-Ubuntu related site. But if there is a way, could someone please explain where Ubuntu stores those packages, and how I save a package for future use?

EDIT: The site finally came back up late Friday afternoon (US time) but thanks to those who replied!

The packages you usually install are .deb files, those files reside on launchpad and could be downloaded as

apt-get download your_package_name

Fully described in this question

In addition to that the installed pachages can be found in

/var/cache/apt/archive

But it won't help you very much, since each package is compiled/build against the libraries of that particular version.

Therefore as maintainer of a package you have to create several "variations" of the same code - check my PPA as example. So taking these packages -made for Ubuntu 20.04 and installing them in 21.10 might succeed, if the libs haven't changed, but you can't count on that. So it is rather futile to "save packages for later use".

So this is where "snap" comes into place - a snap package delivers all the libs needed and is not really dependent on anything outside. The penalty is redundant code, slow performance and not to mention the different look and feel of the UI if you have selected your own theme.

Finally there is a possibility to create an LXC container (lighter than docker) with the correct Ubuntu version, install your app there and share your native X-Server and let that software run in this container.

You have to carefully follow TVHeadend official installation guide from https://tvheadend.org/projects/tvheadend/wiki/AptRepositories .

And then you can get the link to Lauchpad PPA which provides newer actual build of TVHeadend.
The newest package here for your Ubuntu 20.04 LTS was released at 2022-01-13 01:50:21 UTC with 4.3.0~pre+202201122318-0~built202201130132~gitc7b713edb~ubuntu20.04.1 version.

So it is installable by:

sudo apt-add-repository ppa:mamarley/tvheadend-git
sudo apt update
sudo apt install tvheadend

and its packages maybe downloaded directly from launchpad, found at /var/cache/apt/archives after installation or downloaded manually to any folder using apt-get download tvheadend .