in the past they have released $5 tickets 2 hours before game time, available only at the box office. i assume they are still doing that but don't know for sure.
With the $5 tickets, they are uncalpable. You must go directly from purchase to inside Nats Park. So, you cannot just swing by and grab some. As for day of game, sure, every team in every sport does it, but it's risky to count on it. But by all means, to for it.