How to migrate your ArcGIS default geodatabase to an Enterprise Database

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If you use a regular geodatabase in your ArcGIS Server and, for some reason, have to migrate it to an Enterprise Database (such as Oracle, SQL Server, et cetera), you can follow the process below (these steps are for Oracle databases only):

1. Create a standard Oracle database inside a server of your choice.

2. In order to be able to connect to your Oracle database, you will have to install Oracle Client in your ArcGIS environment. If you are having trouble to install it, follow the steps in this link> /support/pages/node/1113093

3. Open the Catalog inside your ArcMap.

4. Expand the Database Connections section.

5. Double-click Add Database Connection.

6. Fill the required fields:
    -Database platform> Oracle
    -Instance> <Database's host address>:<port>/<Database's SID>
    -Authentication type> Database Authentication
    -User Name> sys
    -Password> <your sys's password>

7. Click OK.

8. Open the ArcToolbox inside your ArcMap.

9. Open Data Management Tools.

10. Open Geodatabase Administration.

11. Double-click Create Enterprise Geodatabase.

12. Fill the required fields:
    -Database platform> Oracle
    -Instance> <Database's host address>:<port>/<Database's SID>
    -Operating System Authentication (optional)> <unchecked>
    -Database Administrator (optional)> sys
    -Database Administrator Password (optional)> <your sys's password>
    -Geodatabase Administrator (optional)> sde
    -Geodatabase Administrator Password (optional)> sde
    -Tablespace Name (optional)> <leave empty>
    -Authorization File> <usually located in C:\Program Files (x86)\ESRI\License"xxx"\sysgen>

13. Click OK.

14. Open the Catalog inside your ArcMap.

15. Right click the Folder Connections.

16. Click the Connect to Folder option.

17. Insert the path of the database which contains the data you want migrated and click OK.

18. Open ArcToolbox inside your ArcMap.

19. Open Data Management Tools.

20. Open Geodatabase Administrator.

21. Double-click Create Database User.

22. Fill the required fields:
    -Input Database Connection> <select the connection you created on step 6>
    -Database User> arcgis
    -Database User Password> <whatever password you want>
    -Role> <you can leave this option blank>

23. Click OK.

24. Now you'll have to create a new Database Connection using the user you just created.

25. Delete the Database Connection created on step 6.

26. Inside your Catalog, open the Folder Connections and then the folder you mapped on step 17.

27. Right-click the database which contains the data to be migrated.

28. Go to Export and then click on To Geodatabase (multiple)....

29. Put the Database Connection you created on step 22 inside the Output Geodatabase field and click OK.

30. Wait for the process to finish.

31. Done!

I recommend you to check if everything went fine by looking at the results window of your ArcMap.
You can see more details about that on the following URL:
http://desktop.arcgis.com/en/arcmap/10.3/analyze/executing-tools/using-the-results-window.htm
Also, if you see any exclamation mark on your layers, you can fix that by following the steps on this URL:
http://desktop.arcgis.com/en/arcmap/10.3/map/working-with-layers/repairing-broken-data-links.htm

And that's it! Republish your Map Service and enjoy!