I would recommend to use the native Data Export features in Eloqua, you can define which object you want to export (Contact, Account, Activity), you can also schedule the export e.g. daily.
Data Importing and Exporting
You can also use APIs to get the activity data but if you want to export activities you have to use processed activities as well, more documentations about it:
Hope this helps.
Where would you like to export the data to? That may impact what solution you choose (native or exporter apps that are available on the Oracle Marketplace). Also, one thing I would note is that Contact and Account exports are separate from Activity exports. You may need to set up an export for each depending on what data you are after.
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In addition to what la*16 wrote; also keep in mind
Important: There is a maximum of 5 million records per Activity export sync. If necessary, you'll need to use a date/time filter to reduce the number of records exported to below 5 million. Activity volume can be checked using Insight reports. No filter is needed if there are fewer than 5 million records. Learn more about Bulk filtering.
The steps are available here