Deployment fails due to "queued background process"

This issue can appear for a number of reasons but it's very common when an organization is close to the Custom Fields per Object limit and hence delete a high number of custom fields in one go.

Say, for example, an org has a limit of 100 fields per object and approximately 95 are being used all the time,  but on the same day 10 custom fields are deleted and 15 custom fields are created immediately due to some business requirement. The org will then be hitting the error as the fields you removed won't be hard deleted.

The deployment fails due to insufficient custom field slots being available, when the error is encountered, you will see the following background job(or jobs) being enqueued:

Cleanup of custom field data when a custom field definition is hard deleted

In order to track the progress of these jobs, please use the following Click path:

Lightning:

Setup | Environments | Jobs | Background Jobs

Classic:

Setup | Monitor | Jobs | Background Jobs

Note: you may see more than 1 job associated with "Cleanup of custom field data when a custom field definition is hard deleted", this simply means that custom fields associated with other entities are in the process of being purged

Once all jobs associated with "Cleanup of custom field data when a custom field definition is hard deleted" are complete, please proceed to deploy and you should no longer encounter the error.

Please log a case with Salesforce.com Support should you encounter the error despite the jobs being complete and no additional background jobs have been enqueued.