Does anyone know of public land to shoot on in northern Virginia?

There's a public range on CF Phelps Wildlife Management Area in Fauquier County. Not sure how close that is to you. It's called a "site in" range and is open in the fall to early spring. I am not sure if handguns are allowed.

https://dwr.virginia.gov/wma/cf-phelps/

There is also an ap out there that lets you find private land that owners will let you hunt or shoot on for a reasonable fee. I have a friend that uses it in NC and has locate quite a bit of land to go out and shoot ARs and other rifles. One owner even let then shoot tannerite. I don't use the ap and can't remember the name of it. But research that.

Also, if you live in NOVA, check out Quantico Shooting Club. I've heard good things about it. They have a 1,000 range and I've heard a 2,000 yard range is coming. Even though its on a Marine base, civilians can join. And their costs are way less than my gun club's fees. If I lived up there, I'd be a member in a minute.

https://quanticoshootingclub.com/

These might be a bit West and South of you but here is a link to public land shooting ranges with a lot of information..

http://www.eregulations.com/vi...-s-national-forests/

I just joined the Izaak Walton League in Leesburg.

It is a very nice club with great members.