How can I define an interface for an array of objects with Typescript?

You don't need to use an indexer (since it a bit less typesafe). You have two options :

interface EnumServiceItem {
    id: number; label: string; key: any
}

interface EnumServiceItems extends Array<EnumServiceItem>{}


// Option A 
var result: EnumServiceItem[] = [
    { id: 0, label: 'CId', key: 'contentId' },
    { id: 1, label: 'Modified By', key: 'modifiedBy' },
    { id: 2, label: 'Modified Date', key: 'modified' },
    { id: 3, label: 'Status', key: 'contentStatusId' },
    { id: 4, label: 'Status > Type', key: ['contentStatusId', 'contentTypeId'] },
    { id: 5, label: 'Title', key: 'title' },
    { id: 6, label: 'Type', key: 'contentTypeId' },
    { id: 7, label: 'Type > Status', key: ['contentTypeId', 'contentStatusId'] }
];

// Option B
var result: EnumServiceItems = [
    { id: 0, label: 'CId', key: 'contentId' },
    { id: 1, label: 'Modified By', key: 'modifiedBy' },
    { id: 2, label: 'Modified Date', key: 'modified' },
    { id: 3, label: 'Status', key: 'contentStatusId' },
    { id: 4, label: 'Status > Type', key: ['contentStatusId', 'contentTypeId'] },
    { id: 5, label: 'Title', key: 'title' },
    { id: 6, label: 'Type', key: 'contentTypeId' },
    { id: 7, label: 'Type > Status', key: ['contentTypeId', 'contentStatusId'] }
]

Personally I recommend Option A (simpler migration when you are using classes not interfaces).

You can define an interface with an indexer:

interface EnumServiceGetOrderBy {
    [index: number]: { id: number; label: string; key: any };
}

You can define an interface as array with simply extending the Array interface.

export interface MyInterface extends Array<MyType> { }

With this, any object which implements the MyInterface will need to implement all function calls of arrays and only will be able to store objects with the MyType type.