That's a super interesting question but not a straight up answer. This is what I recommend:
- Use the DC zoning map to determine if the lot is, in fact a lot, or a portion of the public way.
- Also, use DC tax website (and GIS) to determine if there's a previous ownership of the lot
Depending on the tax status of the land, you'll have different options to proceed. I'm not sure if (and how) the city can "sell" public land to a private citizen. I do know that many times, the city hands off public land to NGOs for public uses (like a community farm or park).
Jean Potivien in this group is the master of the city (hahaha, hi jean) and, once you learn more info about the lot itself, he will be able to guide you through it.
Good luck!
ile