I lived in DC for 8 years (recently), and actually worked and lived on Capitol Hill. You can live on the Hill, but it tends to be pretty expensive for what you get, and finding the right area difficult because it's not clear cut in terms of what areas are safest. It's almost block by block in some places.
The center of the Hill is served by three Metro lines...Orange, Blue and Red. The Orange and Blue stop is Capitol South, which is on the House side of the Hill. The Red Line stop is Union Station, which is on the Senate side.
The nearest neighborhood on the Orange and Blue line
that you could consider safe would be Foggy Bottom. It's in DC so you're close to work. It's where George Washington University is located, so it's pretty active most of the day/night...also additional security thanks to GWU police.
Rents there average around $1300 (utilities included) for an efficiency in a security building. It can be kind of loud though because of university stuff.
A lot of newcomers like to find a place in Northern Virginia, just across the DC line, near a Metro stop. I did that when I first moved there and quickly regretted it. If you want to do anything on your day off or go out, you'll find yourself in DC most likely so why not live there. There are plenty of safe neighborhoods.
My favorite neighborhood in DC (it's on the Red Line), is Cleveland Park. It's like a small town in the city. A really beautiful area north of downtown. It's pretty vibrant...young people, younger couples, white collar professionals. The area around the metro stop (3 blocks long) has pretty much everything you could want...restaurants, bars, grocery stores, wine/liquor store, health club, coffee shops, dry cleaners, drug store, an amazing art deco style movie theatre...you name it. Your commute on the Metro would be about 20-25 minutes, including your walk to the Cleveland Park station and from the Union Station stop to work.
I would highly recommend looking for a share, versus trying to go it alone. Rents in DC are pretty expensive, so sharing gets you more for the money in terms of quality of housing...also its a good way to meet people there. Check out Craigslist for shares.
I attached some Cleveland Park shares, and I think you'll see some better examples versus going it alone. Good luck!