No, it is generally not possible. The dealbreaker problem is not with Android, but rather that just about any mobile network provider is going to block incoming connections.
For special purposes, you can do things like have the phone contact a publicly visible remote machine and create a tunnel with something like ssh through which inbound connections can then be sent if there is a server program listening on the loopback interface of the Android device. But this tends to be only useful for things like remote testing.
Were you using wifi you could do this, but you'd need to accept that Android is not designed to promise a high availability of any 3rd-party program. You would need a wakelock to keep the processor running. And you may even need to do things like periodically initiate outbound traffic from the device in order to keep the wifi stack/radio in a fully active mode so that it can accept connections.
Generally speaking, data you want to make remotely available from the device should be proactively uploaded to a mutually visible server. If you want to send requests to the device, look at using something like Google Cloud Messaging which is designed to wake up the device and a recipient application on it in order to deliver a brief message.