Assuming I am: a- reasonably intelligent b- familiar with Word, Powerpoint, and Excel c- aware of what a database is supposed to do
Is there any reason why I can't, for example, set up a very small database of contacts' phone numbers and addresses, pratice searching and sorting, and then apply for jobs that say they want me to be able to use Access?
What else does one do with a database? Oh, mail-merge. What else should I find out how to do before I claim I can use it?
It depends on the level of expertise that employers are asking for tbh. They might want someone who can use an in-house database that has already been set up ('use' could mean everything from basic data entry to setting up complex queries), or they might want someone who can create one or more databases which could be anything from a basic contact list to something considerably more complex than that. Anyway, having said all that, I'd say that there's no reason why you can't claim to be able to use Access if you can set up a simple database and manipulate the data in various ways. If you're fairly IT savvy and can use the other Office apps I'm sure you won't have any problems at all in picking it up!
Data modelling (including entity-relationship diagrams or similar -- evidence of understanding one-to-many and many-to many relationships, at any rate), forms, queries (including SQL) and reporting, probably (depends what they mean by "use it", which may vary from "create your own databases" to "type numbers into this form". But if I advertised that I wanted someone to be able to use Access then that's what I'd be looking for. Being able to use VBA would be a help but is stretching more into programmer territory).
If you are going to create your own practice database you should make sure it has two or three linked tables with appropriate data entry forms (including text boxes, checkboxes, radio buttons, drop-down lists) and that you can query and run reports across multiple tables.
If you get a book on relational databases (look for something with that in the title) out of the library that should give you the theoretical background and then you can play around with Access.