I am stuck on a really old version of Ubuntu, 16.04.
The machine is in an environment with no access to the internet - which explains why it's stuck on 16.

The question is, how many series up upgrades do I need to do to get to latest-greatest 22.04?

Whatever bundle I need, I need to burn it on a DVD or Bluray disk (flash may not be an option, not sure).

Would it be easier to download the ISO image for 22.04 and do a fresh install? I am not sure what is on it and if it's harder to reinstall software versus doing several upgrades.

You do a fresh install; that many versions behind would be a nightmare to update, and you'd probably have to do NUMEROUS updates to even get to the point where 22 would work. Make backups, and plan your downtime. Format/install/reload is your only option, if you want things to actually work right.

If I may ask. Is there a reason to upgrade if its not connected to the internet? Not that I recommend out of date software but if there is almost no attack vector then why fix what isn't broken? Or are you wanting to use a newer package for your service(s) that Xenial won't run?

Nope. let it run... Unless you have a new feature you'd like to use, that isn't available with the old. Way back we had a work Linux file server run for 7 years before it was retired. It was either running Red Hat 5 or 7, (don't remember).

And from 16 to 22 ... Backup data. Install.

Ubuntu allows upgrades from one LTS release to another so from 16.04 > 18.04 > 20.04 > 22.04. Your big problem is that you are 3 months late as support for 18.04 ended in April of this year. It would be a difficult process even for an experienced user as none of the files needed for 18.04 are located on the same server as they were so you would need to find the archives and change your configuration files to access them.

Much simpler to continue using it offline or reinstall 22.04.

Understood
The machine is off-line so any upgrade / patching will be really problematic