How to backup an Oracle database from SQL Developer
No. And then yes. Well, it depends.

What are your requirements? 

Backup / Restore Capabilities

Do you want a backup of the database ...


... for a restore  
... to perform recovery  
... to roll forward additional archive logs  


Then RMAN is your tool of choice. Yes, RMAN is pretty complex and has a steep learning curve. However, once you have mastered it, you will benefit from a large variety of possibilities that RMAN provides you with, e.g. central RMAN catalog for multiple Oracle instances.

Export / Import Capabilities

Do you want a dump of a database ....


... for a "restore" to a new server/instance
... no recovery required
... no additional roll forward of archive logs 


...then you might want to consider using the EXPDP (see: Data Pump Export) and IMPDP (see: Data Pump Import). These tools provide you with the ability to obtain schema-specific data and import the data into different tablespaces; or to filter the export via table names, dependencies. On import the schema can then be re-linked to a different schema (see: REMAP_SCHEMA), etc.

SQL Developer

SQL Developer has the capability to export data using a wizard. This is similar to the built-in EXPDP and IMPDP tools provided with the RDBMS installation. A quick guide to using the wizard can be found at: Exporting and Importing Metadata and Data and there in the section Using SQL Developer for Exporting and Importing (Tools | Database Export).