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Yup, it's possible.  I've worked in the past with a server setup as you describe running six AGs containing around 550 DBs between them...

You can see some of the information by running:

DBCC DBINFO( <<dbname>> ) WITH tableresults

That'll show (among other things) last backup & last log backup...

In msdb, look at tables named "backup<<something>>", eg "backupset".  This is where the information relating to backups is stored.  It's all perfectly readable using normal T-SQL queries.  There's

I'm on an estate where we have ten single-core servers, some with as little as 1GB RAM, and none with more than 8GB...  They are, though, "sized to the workload"...