Yes and no. Yes many of the ruins of Ur still exist, however there has not been any excavations done there since 1930's by Sir Leonard Woolley and much (about 70-85%) of the site has still not been excavated. The main attraction of the site, the Ziggurat of Ur, has been completely uncovered however. After the US invasion of Iraq, the US military guarded the site, and there was not a lot of public access to the site. The US returned control of Ur over to the Iraqis in May of 2009 and a joint excavation with the University of Pennsylvania and Iraqi archaeologists is planned later this year, though I am not sure if any public access is allowed.