Cannot find the chest while doing the Break of Dawn quest

I found the Statue to Meridia, then went to find the chest, but here isn't a chest where the map marker points to: it is inside the Dawnstar Sanctuary for the Dark Brotherhood. Is there any way to fix this?

It sounds an awful lot like this known bug:

Meridia's beacon may spawn in the Dawnstar Sanctuary, even if the Dark Brotherhood questline has been completed, making the chest containing the beacon inaccessible. In turn this forfeits the Oblivion Walker achievement and makes it impossible to finish the quest.

I suggest loading an older saved game and continuing from there.

Alternatively, you can use the console command player.addItem 0004E4E6 1 to add Meridia's Beacon to your inventory. Now you should be able to enter the temple.