Bashir says he needs a runabout to go to Vulcan to get medical supplies for a field hospital he is setting up in a week. Throughout the series it has been established that it takes several weeks to travel from Bajor to the core Federation worlds, characters have complained several times about the long trip. According to Memory Alpha, Bajor is 52 light years from Earth and Vulcan is 36 light years. The Defiant's top speed is warp 9.5 (1,816c), even at that speed it would take roughly 10 days to travel to Earth and seven days to Vulcan, assuming that speed could be maintained (which it can't). However a Danube-class runabout is far slower, it only has a top speed of warp 5 (213.75c), it would take roughly 89 days to travel to Earth and 61 days to Vulcan. So there's no way Bashir would get to Vulcan and back in a week, round trip would be 122 days or about seven weeks.
Commander Sisko folds his arm while trying to convince Gul Dukat that he wants to help disable the Defiant. We see him fold his arms as he says "Then bring me with you..." He leaves his arms folded for several moments as he continues his argument. The camera leaves him for a moment and then when the shot returns to him, we see him fold his arms again as he says "I want to to try and disable it first".
In the first episode of this season, we see the new style combadge be released. This episode aired at the same time of Star Trek: The Next Generation (1987) but the combadges never changed during that series. Thomas Riker is seen in the TNG era uniform with the new combadge but should be with the previous style combadge since the Enterprise crew never updated theirs. This also shows the in-continuity between the different types of uniforms used that has never been explained.
Naval vessels, even when in port, always have personnel aboard them. However, Thomas Riker manages to steal the Defiant with ease due to no crew being aboard to stop him.