I am looking for an 'out of box' software program that will allow me
to create my own customized online membership database to manage
approximately 5,000 members.

I need a software program to do this that will meet the following requirements. 

The database must:
-be web-based on a standard secure web server
-be compatible with MAC and Windows users
-not require computer language knowledge (ie. JAVA, html, etc.) to
create/modify/maintain database
-be customizable to integrate into existing website skin
-generate reports
-automatically generate unique membership numbers
-create expiration dates based on date of membership number creation
-support different membership subscriber types
-support members paying membership dues
-support users managing their profiles
-support tiered access (restricted to some & not others)
-support multiple people (approx. 10) accessing/updating the system simultaneously 
-include an administrator module to manage members status 

Thanks!

If you don't find a suitable one ready-made a decent programmer should
be able to do that in a month - possibly excluding the payment aspects
which I've never worked on.

Agreed. Payment aspects would not even be hard. Simply tie paypal.com
into the website for free. Accepting payments can be free as well
provided you do not wish to accept credit cards. If you choose to
accept credit cards each transaction through paypal would cost no more
than 2.9% of the transaction amount plus .30 cents, per transaction. I
am currently building an online storefront using this approach with a
backend SQL database.