Do I need a RDBMS for building an Across application?

Although a RDBMS is not strictly required by Across Framework, you will need one if you are going to use installers. The RDBMS is necessary for Across to be able to track the the installer state between application starts.

Across itself creates only a minimal set of tables and will do so only when required, that is when installers need executing.

See the reference documentation section on Installers for more details.