C#: Is there a way to classify enums?

You can use attributes:

public enum Position
{
    [OffensivePosition]
    Quarterback,
    [OffensivePosition]
    Runningback,
    [DefensivePosition]
    DefensiveEnd,
    [DefensivePosition]
    Linebacker
};

And then check for IsDefined on an appropriate FieldInfo. Syntax is not very pretty, but you can throw in a couple of extension methods to make things more manageble:

public static bool IsOffensivePosition(PositionType pt)
{
    return typeof(PositionType).GetField(Enum.GetName(typeof(PositionType), pt)).
        IsDefined(typeof(OffensivePositionAttribute), false);
}

You can declare the enums in a class:

public class Position
{
    public enum Offensive { Quarterback = 1, RunningBack }
    public enum Defensive { DefensiveEnd = 10, LineBacker }
}

Note that the Defensive values start at 10 so that values don't overlap. You don't state why you want to do this, so this might not meet your needs.