I spent a decade or so living in Arlington and Alexandria and I wound up not riding at all during that time, for various reasons, but a big one is that it was very hard to find any barn giving lessons within a reasonable to me drive (and harder to find ones within my budget).
Depending on where you are in Arlington, there might be some barns in southern Maryland, on the other side of the Potomac that are reasonable distance (I feel like Willowbend was this for a lot of people, but unfortunately it was sold to developers a couple of years ago).
I’m in Maryland, and at least an hour from DC, so I don’t know a ton of barns in your area. I have participated in catch ride shows at Twin Oaks (lovettsville in Loudon county, probably a hike for you, and more eventing focused, but the horses were lovely and I’ve enjoyed every rider I’ve met from there). And I actively enjoy the coach (she’s judged some scrimmages we’ve had at other barns)
Also participated in a catch ride show at Tamarack Stables in Lorton. The horses I was on were lovely, other horses were not so great. So they might be hit or miss, but they’re an option.
If you’re far enough north in Arlington, or have a flexible schedule, there are a lot of barns around poolesville in maryland. But that’s probably 30-45 minutes from you without traffic. Granted, it’s been over a decade since I lived in NoVA, but traffic was hard then and I doubt it’s gotten better since.