The exception is the main post office at 2 Mass Ave NE, which is (blessedly) around the corner from Union Station, and thus Metro-accessible. Today at least, they were open until 3 pm (with very few customers, considering). They expect to be open tomorrow as well.
This has been a huge nightmare for me as I have an eBay business and this afternoon I went to a couple of Dupont-area POs to drop off nearly 40 (thankfully lightweight) packages and found both offices closed. I had the presence of mind to call the main PO and rushed over on Metro before they closed too. USPS.com has a page of "updated daily" weather delays/closing notices, but it is clearly not updated daily, as there is no mention of DC whatsoever. Just thought I would let y'all know!