[First in an occasional series of FAQs about traveling to Washington, D.C. For more, please click on the "dc-faqs" tag above.]
Left-luggage facilities are fiendishly difficult to find in the USA. Security overkill accounts for some of that, sure, but countries with far more experience with train-station terrorism still have plenty of luggage lockers.
First up: it can’t be repeated enough, but pack light. Not having to wrestle with lots of luggage means much greater flexibility. Carry as little as possible while sightseeing in Washington, since airport-style security checks are commonplace. In particular, keep metal objects to a minimum.
Small bags can be checked at several Smithsonian museums; just be sure to remove anything that might raise a security guard’s eyebrow, and queue up to retrieve them before the closing-hour rush. Note that Gallery Place has lockers and closes at 7PM, not 5PM like most of the other museums.
If you’re departing or arriving by train and don’t need your bags overnight, Amtrak allows passengers to check bags in a day before, or to pick up checked bags long after arrival. The key caveats: relatively few stations have baggage service, and on the Northeast Corridor, bags only go on the overnight train (so you’ll have to check your bags the night before or retrieve them the day after).
Similarly, you can usually check bags for airline flights anytime on the day of travel. This would be a tremendous time-suck in most cities, but it’s easily done for flights from nearby DCA.
Hotel bellhops check bags for arriving/departing guests as a courtesy, and if you play your cards right you can usually take advantage of this. Dress like a business traveler, tip generously, and don’t lie if asked. You’re best off trying large convention hotels where the bell desk is outside the lobby (not smaller hotels where it’s at the front desk); walk into a side door of the hotel and out the front door to the bell desk.
One last resort is the left-luggage facility at Union Station. It’s expensive, but it’s also right there.