Can a tenant without a lease be evicted in 30 days?

No.  As long as you haven't broken the unwritten lease (you've paid the rent, haven't damaged the property or anything like that), you can't be evicted in less than 90 days, for the landlord's personal use.

You can see an internet copy of the DC government's tenants rights brochure, with a lot of information and resources, here:   http://dcra.dc.gov/dcra/lib/dcra/information/hra/eviction_brochure.pdf

As you'll see in the brochure, you're lucky to be living in a city with probably more law students per resident than anywhere else, and quite a few of them work in landlord-tenant clinics to get courtroom experience under close supervision, where you might be able to get represented for free.