No. You do not need an internal Hard Drive to play music on Xbox 360.
Supported Music Sources (No HDD Required)
USB Flash Drive / External HDD: Format as FAT32 (Max 32GB per partition on 360, or use PC to format larger FAT32/exFAT → 360 reads FAT32 only). Copy MP3/WMA/AAC files. Plug into Front/Back USB. Access via Music Player / Guide → Music
Audio CD: Insert Disc → Play / Rip to HDD/USB (if HDD present)
Media Server (DLNA): Stream from PC (Windows Media Player "Stream" / Plex / Serviio) over Ethernet/Wi-Fi. System → Network Settings → Media Sharing
iPod / Zune / MP3 Player: Sync Cable → USB Port. Appears as Portable Device
Xbox Live Marketplace / Zune Music Pass (Legacy): Streaming (Requires Gold/Internet)
Benefits of Internal HDD (20/60/120/250/320/500GB)
Rip CDs: Store library permanently (WMA/MP3)
Game Installs: Faster loads, less disc noise (Disc still required for auth)
Profile/Saves/DLC: No Memory Unit needed
Xbox Originals / Arcade Games: Require HDD (Most)
Recommendation: A 64GB-128GB USB 3.0 Flash Drive (Formatted FAT32) is the cheapest/easiest music solution for a 4GB/250MB Slim/E model.
