In Object Mode, I can type in xyz coordinates using this menu:
What is the equivalent in Edit Mode, so I can precisely position the parts of an object?
Both in Object Mode and in Edit Mode, you can use the Transform menu of the Properties Panel on the right-hand side of the 3D View (shortcut N).
If you have only one vertex selected, it will position that vertex precisely. If you have more than one selected, it will affect the median position of all the selected vertices.
Edit Mode:
Object Mode:
N
panel on the right of 3D View because it allows entering values for Object mode when in Object mode and for Edit mode respectively. $\endgroup$