it just feels wrong to have a class that does nothing but validate and store values
For me, it feels perfectly fine - as long as you can come up with a meaningful name for the class. I don't know your domain, but Angles sounds pretty meaningless to me - but maybe it is just so inexpressive for the sake of this example, not in the real code. Same holds for convert - give the function a name which expresses clearly what kind of conversion it does - from what input to what output. And the comment behind _Angle0 indicates the method should have a name like _validateAngle0, for example.
Alternatively, consider to implement the conversion functions as class methods of Angle. It is probably not possible to implement them as different Angle constructors, since they are cannot be distinguished just by their signature, so class methods should be the canonical choice here.