Joe Sam Shirah
Few companies can really afford NOT to use a DBMS. If nothing else, developer time spent on custom persistence methods can quickly surpass the cost of a DBMS. If the company doesn't understand this, one option often offered in jest may get serious: Find another company.
But there are low cost, production quality options available and it's worth using the drivers list ( see Where can I find a comprehensive list of JDBC drivers, including the databases they support? ) in reverse to check out DBMS candidates.
Open source DBMS engines are another possibility. For some options there, see Tools for the Open-Source Database
Another, possibly larger, consideration for a cost concious company is that a number of the large commercial databases effectively require a full time DBA, meaning another salary. For that reason, you'll want to look for an easy to use, low maintenance product.