MacOS Ventura 13.3.1 (22E261) can only access internet for 5 seconds after rebooting.

Try opening one such location while running in Safe Mode.

Does it work? Then you have some software conflict.

Do you have any antivirus? Any vpn software?

Please run Etrecheck and post its full report here. Use the "additional text" button and paste the report into the text box.

Your battery is failing, and that can cause all sorts of instability in your system.

Your free space is getting a bit low, at just over 10%, but I think the battery is something that you need to address.