There are some common things for start-ups, one of them is being tight on money. As a result, they will opt for using open source technologies available, rather than paying licenses for some big company.
For the database side, that means they will most likely stay away from Oracle/Microsoft stuff.
Secondly, DBA is a very specific area of expertise. For most start ups, dealing with the quirks of database/optimizing stuff in not high on the priority list, thus they won't be needing a DBA at early stages. That generally rules out startups that are in early stages.
Consultancy can be an option. But in my experience, most startups will avoid paid consultancy, as they won't have that much money.
So, in short, it will be hard to find a start up, especially in its early stages, that needs a DBA, especially a DBA specialized in Oracle/Microsoft products. Not impossible, but hard.