Can you shut down an account due to inactivity?

Since accounts can be used across multiple Microsoft services, there's no way to know exactly how inactive the account actually is. If they don't sign into the Xbox account within 5 years, the tag will be freed up. If they don't sign into their Microsoft account within 2 years, the account will be closed which should also free up the tag. There are also multiple accounts using that tag already, so even if the original account is freed up, you would be given a suffix since there are other accounts with the same tag.

Even the PC Marketplace is currently shut down and inactive due to some concerns from the PUBG folks. Not sure what its status will be but its ultimate fate is likely to be the same on Xbox as well.

You don't have it set to shut down after an hour of inactivity do you?

No. The tag system is in use on other Microsoft systems as well. So even if they aren't using the tag for Xbox, it could be in use elsewhere.

If you haven't logged into that account in a long time, it might be gone. I believe that it is 2yrs of inactivity for a Microsoft email to get deleted and possibly 5yrs for an Xbox account to be shut down. You might be able to see the account when searching for it but it will no longer be accessible.