Why does my xbox 360 keep disconnecting from Xbox Live?

You can troubleshoot your console's internet connection from the settings menu. This will tell you if it is caused by a modem issue, hardware issue, or if the Xbox Live services are unavailable. Right now this post does not offer enough information to determine why you're having intermittent connections but the likeliest cause is simply the age of the console. At more than ten years old, your Xbox 360 may be experiencing hardware failure that prevent it from connecting to the internet.

You can refer to Microsoft's support page for further help fixing the problem: https://support.xbox.com/help/hardware-network/connect-network/xbox-360-network-connection