Jews had been discriminated against for more than 600 years by the time, so both the discrimination they endured and the rumors of further persecutions were nothing new, and Jews had always survived them in the end. This is why Elie's family did not seek asylum in another nation, that, and the fact that no one wanted to believe how bad things could get, so they constantly reassured themselves that they could avoid such a horrible fate.
The people of Sighet had heard rumblings, so there would have been an opportunity to escape, but no one believed the rumblings. Elie asked his father to escape, but that idea was turned down. Moishe the Beadle tried to warn all of what he saw, but no one paid heed. The were too complacent, unbelieving, or compliant.
Good points presented above. Elie and his family have several other opportunities to leave before they are taken away by the Germans. One is when their servant leaves and offers to let the family come with her. They (or should I say he, meaning Elie's father) did not take advantage of that opportunity. Another opportunity was presented when Moshe the Beadle came back to warn them all of their impending doom; however, no one believed him and no one left. One final opportunity came when they lived in the ghetto and the back of their house was accessible to the "outside," someone knocked on the window, presumably to offer them a chance of some kind to escape. They do not, of course, take advantage of that opportunity either.
Well, there was really only one option for escape: leave the country. However, very few families took this chance. Many did not believe the rumors, even when individuals returned and told stories of mass killings and deportations. Once the Fascists gained power in Hungary, some were concerned, but most citizens considered it just another shift in administration. At one point, Eliezer asks his father to leave, but Chlomo Wiesel claims he is too old to start again. Thus, the Wiesel family does not take the chance to escape while they have time. Finally, when the Germans actually move into the town, they are quite friendly and polite, and thus no one really considered what was coming.