Dining Near Arena Stage
I live near there and the options are woefully limited. For a finer dining experience, you have Sou'Wester and CityZen in the Mandarin Oriental, but it's a pretty long walk from there to Arena Stage. Along Water Street are Jenny's (asian), Phillips, Cantina Marina (basically bar food) and the bar/restaurant at Channel Inn (never eaten there, but the bar is fun!). There's a Subway on 4th right across from the Safeway. Seriously, we can only hope that the Arena Stage renovation and the revitalization of the waterfront brings more options down there.
My advice would be to get there very early, use the $1 circulator bus (stops right near Arena Stage) to go up 7th street to places like Oyamel, Rasika, Jaleo, Zaytinya, etc. Walk over to 9th to get the circulator back to the theater. The circulator bus runs every 10 minutes.
Alternatively, the theater is a short walk from the Waterfront Metro (green line), so you could eat at any place on the green line and then metro to the theater.
1330 Maryland Avenue SW, Washington, DC 20024
633 D Street, NW, Washington, DC 20004
701 9th St NW, Washington, DC 20001
600 Water St SW, Washington, DC 20024
1101 Sixth Street, SW, Washington, DC 20024