The Byzantines had been stripped of virtually all of their territory and power by this time and the empire was little more than the city, the Prince's Islands in the Sea of Marmara, and the Peloponnese. The city itself was in relatively good shape as the walls had recently been repaired under the previous emperor and the men inside felt confident they could hold out until relief from the West arrived. The force inside the walls totaled around 7-10,000 soldiers opposing an Ottoman force of around 60,000 men. Despite offers to Constantine XI to simply abandon the city and surrender with dignity the Byzantines refused believing help would arrive to drive off the Ottomans. In the face of such fierce resistance the Ottomans became divided in war councils as to whether to continue or break off the siege. Mehmed decided on a final all out assault that eventually broke the defenses at numerous points and positions. When an Ottoman flag was raised over the postern gate Kerkoporta it caused mass panic among the defenders who abandoned their position to defend their families or flee the city. Constantine XI and a small cadre of loyal bodyguards perished in the street fighting as the Ottomans poured into the city. Mehmed allowed his soldiers to pillage and loot the city for three days exempting important government or cultural centers which he placed soldiers to protect. Constantinople's capture and Constantine's XI death marked the end of the Byzantine Empire.

TL;DR The Byzantines were prepared for the siege, but lacked the military power to turn away the Ottomans as they had before and a lack of assistance from Western kingdoms doomed them to their fate.