This error generally comes when you lost connection with SFDC server due to network failure or your log is not available to show right now. I can notice in your screenshot that you have multiple log opened which are 2-3 days old, they doesn't exist on SFDC server now that is why you are seeing this error. Try to close all existing open log and start your developer console again. Developer console will definite open.
Thanks, Himanshu Salesforce Certified Developer, Administrator, Service Cloud Consultant
P.S. If my answer helps you to solve your problem please mark it as best answer. It will help other to find best answer.
This error generally comes when you lost connection with SFDC server due to network failure or your log is not available to show right now. I can notice in your screenshot that you have multiple log opened which are 2-3 days old, they doesn't exist on SFDC server now that is why you are seeing this error. Try to close all existing open log and start your developer console again. Developer console will definite open.
Thanks,
Himanshu
Salesforce Certified Developer, Administrator, Service Cloud Consultant
P.S. If my answer helps you to solve your problem please mark it as best answer. It will help other to find best answer.