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Yes, delivering for Doordash is self-employment. If you deliver for Doordash in the United States, you do so an an independent contractor. 

There's some controversy around whether Dashers are misclassified, or whether Doordash is illegally using independent contractors. In late 2022, the Department of Labor has proposed a new rule that examines an employment relationship. Doordash would likely fail the test under that new independent contractor rule.

However, all of that controversy has yet to play out. Until then, Doordash Dashers are contractually self-employed.