As a super admin, it's easy to export all the data for my business. You say that the admin account here is long gone, but every business needs to have one. I'm assuming that this is a legacy free account, otherwise there would be someone paying the fee monthly who would be able to contact Google to reset it. If so, you have a short window to get the information archived, because it looks like Google is still planning to make changes to legacy accounts
starting the end of June.
If you have any suspicion that the account credentials might belong to a super admin, you can check easily by clicking on the profile image in the upper right portion of the Gmail interface. If the account belongs to an admin, there will be a link to the Google admin console. Even if you can't access it, you could probably request a new password at https://accounts.google.com/signin/recovery
assuming you have the right admin email or recover email information. Without that, though, you apparently need to be able to modify the DNS for the domain.
IMAP and POP are also enabled or disabled by the account admin, though again I think they default to on. There's a further restriction possible in terms of IMAP clients -- it can be set to any mail client or be restricted to specific clients using OAuth. IMAP is probably still your best option, though, if indeed there is no super admin user.
Danielle Cunniff Plumer
Texas State Library and Archives Commission