As is the case with many time travel stories, Earth travels through space as it travels through time. It's much more likely Sisko and Bashir would have materialized in empty space, rather than on Earth directly below the Defiant's coordinates but ~350 years in the past.
Bashir attempts CPR on Gabriel Bell to treat a stab wound to the stomach.
When Sisko and Bashir fight the "ghosts," a number of the punches thrown clearly miss.
As the Defiant orbits Earth just before Sisko and the others beam down, a small protrusion can be seen sticking off the bottom of the port warp nacelle. This must be a piece of filming equipment used on the Defiant model, as it is not part of the ship's design and is not seen in other shots.
Sisko, Bashir and Dax are in San Francisco from late August to early September, yet everyone is wearing layers of clothing (coats, sweaters, etc.) and gathered around fire barrels to keep warm. It is towards the end of Summer at this time of the year.
Sisko brings Bashir breakfast from the food lines. Either he was somehow able to obtain two meals (even if he had Bashir's food card, it is unlikely he would be given two sets of rations himself, given the state of the sanctuary district) or he gave Bashir his rations and went unnecessarily without, which would be poor form from a seasoned officer (the same reason you affix your own oxygen mask in a crisis before helping others would apply here).
As with all Star Trek that goes into earths past they seem to pass over the subject of the Eugeneics wars. Is this not the time period. Are these problems also a result of augmented biological humans being in control?
When Vin logs into the fingerprint I.D. system, the voice prompt tells him that his government discount has been applied. A low level employee would not be required to pay to process new arrivals as part of their job. If they were, that would disincentivize them to bring in transient people, which Vin is clearly eager to do.
Sisko, Dax, and Bashir beam down to attend a formal event, which O'Brien has stated requires dress uniforms. Yet they are wearing their regular duty uniforms and did not carry their dress uniforms with them to change into.
Julian says that if they ever get back to the station he promises not to complain about its Cardassian beds ever again. The beds on DS9 were supplied by Starfleet, as established in dialogue in the pilot.
Some of the residents of the sanctuary district trade their clothing for Sisko and Bashir's Starfleet uniforms. The clothing they trade away includes heavy winter coats, and it is shown to get cold at night. The previous night, Bashir and Sisko are shown trying to keep warm, indicating their their uniforms do not provide good protection against the cold. It would make no sense for a resident of the district to trade away a heavy coat even if the uniforms they receive in return are cleaner.
Bashir tells Sisko that 21st century history isn't a hobby of his, as it's "too depressing." First Contact with Vulcans takes place in 2063 and work begins towards Earth becoming the near-utopian society seen in the Federation and Starfleet. While the first half of the 21st century is marred by war on Earth, the latter half is far from depressing.