You didn't indicate if this server worked previously and is now not working, or if you've always had this problem.
If this is a new problem, have you looked into the activity log to see when/why your services are shutting down? Is it all of them, or just one? There should be some troubleshooting clues there. You can adjust how much info is logged from the Server Admin program, Admin|configuration|Log.
If this is a problem that has existed all along, or just recurs after rebooting the server, you need to check the startup options for the services to set your engine services to "automatic" startup. This is not done by default in the ARCserve installation process.
This is a behaviour you will see under the default installation, right after a reboot of the server. ARCserve will not start the services after a reboot unless you set them to "automatic" in the services control. They will be started when ARCserve Manager is run, if it is set to "Autostart Engines" under the file menu, which is also set by default.