Can You Connect an Xbox 360 to the Internet via Dial-Up?
No. The Xbox 360 does not support dial-up connections.
Technical Reasons
Bandwidth Requirement: Xbox Live and system updates require broadband (minimum 512 Kbps down / 128 Kbps up recommended). Dial-up (56 Kbps max) is orders of magnitude too slow.
Hardware Incompatibility: The console has an Ethernet port (and Wi-Fi on later models) but no RJ-11 modem port. The network stack lacks PPP/dialer software.
Authentication/Latency: High latency and frequent disconnects inherent to dial-up break the persistent connection Xbox Live requires.
Alternatives for Dial-Up Users
Upgrade ISP: DSL, Cable, Fiber, or Fixed Wireless are the only viable paths for Xbox 360 online.
Offline Play: Single-player campaigns, local multiplayer, and LAN System Link work without internet.
Use a PC: A computer can connect via dial-up for basic browsing/email, but cannot bridge that connection to the Xbox 360 effectively.
