Do I need to file Self-Employed?

Yes

If this is reported on 1099-NEC, that is Non-employee Compensation and you have to file it.

It is considered Self-Employment income. 

You'll also need to report the vehicle you used which may result in Depreciation recapture.  

I didn't receive a 1099-NEC and have no way of knowing if DoorDash filed anything on me. What now?

Businesses are required to file a Form 1099-NEC for anyone who was paid at least $600 during the year. DoorDash was not required to send this form to you.

Self-employed people who earn less than $400 a year are under the threshold that requires you to pay self-employment tax. If you had other income in addition to the DoorDash income you are required to report the other income and include the money from DoorDash.

@Odomj 

@PattiFso just to make sure, I earned about $10,700 working other jobs, and because of this I do need to report the $250 I earned through DoorDash?

Since you had other income, you need to include the $250 of self employment income. If you did not have any expenses related to the job (if you didn't use a vehicle but walked), you would not be automatically upgraded to TurboTax Self-employed.