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Look into Custom Object Programs. These are programs that can give you some data manipulation tools for custom objects.
Note that they would delete the entire custom object record, not just a value from one of its fields. So I am not sure if I understood well your case, but you can test and dig deeper into it.
For e.g. you can first create a normal program (for contacts) use a segment to select contacts linked to your custom object which you want to delete (or 'Compare Custom Object Fields' step in program canvas) and route your contacts to a custom object program with 'Add Linked Record to Program' step. Then you can use 'compare custom object field' and 'delete custom object record' steps to select and delete the desired records:
E.g.: If you contact is linked to a custom object called 'campaign member', your contact will also probably be part of multiple campaigns - so it will have multiple records:
When you add your linked records from the contact program you will essentially add all your campaign member records linked to the contact. Then in the custom object program, you want to ensure you use the 'compare custom object fields' step to select only the relevant records to be deleted.
If this sounds like a solution for your case, I'd suggest to first perform some tests on dummy records/contacts to ensure you understand how it works. Once they are deleted from Eloqua, they are gone. So a backup would also help.
You can also find other suggestions here: Backup / Export Custom Data Objects
Hope it helps.
This is not possible and should be moved to Dream It. I will definitely vote this up. Deleting CDO field values one at a time, or uploading null values is such a hassle.