The title pretty much says it all. They were evacuating a military base under attack to save their lives, and they thought that they were evacuating to Earth. Who would carry things like a shaver, a needle, gym clothes or who-knows-what-else? How come they have such things aboard Destiny? I doubt they would be present on the ship itself...
Did the creators of the show simply ignore this nonsense?
During the Pilot episode, as they evacuate the base you can see a number of crates and boxes of supplies come through the gate behind them.
I think we can assume that these boxes contain your mystery shaver and various other fripperies such as food.
This is backed up by the episode's official novelisation (which was itself based on the show script)
Young didn’t press the point and turned back to the lieutenant. Make sure that everyone carries as much of the expedition supplies as they can.
Scott gave him a questioning look. Why?
You’re not going to Earth. Rush dialed the ninth chevron.
and
Young saw the chaos at the mouth of the gate and shouted at the top of his lungs, snarling at the evacuees to get through as quickly as they could. The gate room and Icarus itself was coming apart as chunks of concrete and rebar fell from the ceiling, smashing through gantries where they landed. Control panels sparked and dimmed as power began to die off through the base’s crippled systems, and the colonel knew that the death knell was only seconds away.
Greer was almost at the event horizon, supporting an injured civilian with his free hand. He caught sight of his commander and called out. Sir!
Young stabbed a finger toward the Stargate. Go! he shouted, and reluctantly the Marine obeyed.
Dozens of supply cases and backpacks were scattered across the floor, left behind by the fleeing evacuees, some too heavy to carry, others broken open with their contents scattered. He looked up to see the last few people passing through the gate, and Young halted, instead turning to the crates and starting a frantic search.
