And a movie theater.

That's actually not true. Vouchers are legally viewed as a source of income, which cannot be discriminated against in Washington DC. Now, do some landlords still do that? Sadly yes. And they can be taken to court by the applicant for some good money and/or accommodation if the applicant is wise to the law.
Landlords and property managers are required, by law, to accept vouchers.
That being said, DCHA (now split into two organizations) was so messed up that the application process would get hung up for months, during which time another person with money ready to go can walk in and apply to take the property. The landlord can continue to list the property until payment is received by an applicant, and it's easy to see arguments on both sides for discriminatory behavior.
It's all a mess.

It depends on the place. Many places, like the ones I manage, have listings under $1k due to rent control. At the very least you can always inquire. Just be sure not to give money up front, by any means, without taking a tour!