You can proceed in two steps: rotate the object the angle you want, then move it along its axis.
For a better visualization, you can activate "show axis", here we want to move the monkey from the cube:
From that rotate the angle you want: R and use the mouse and eventually set the value manually in the operator panl.
Then move along the axis G then XX (twice) to move along local axis, and again you can also set the value manually.
To keep them in the same configuration when moving, select one then shift select the second and use CtrlP.
You can just use a Limit Distance constraint for that. Simply add the constraint in the Constraints properties and set the target to your other object. Set Distance to the distance you require and Clamp Region to 'On Surface'. This will constrain your object to remain on the surface of a sphere projected around the target object.