I think you're probably fine, looking around it sounds like it's just having trouble connecting to your router. See if this works.
A PS3 Error 80710723 signifies a network/router error. Try going to SETTINGS > NETWORK SETTINGS > INERNET CONNECTION SETTINGS > (if wired, choose path 1. if wireless, choose path 2.)
1. WIRED CONNECTION > CUSTOM > MANUAL SETTINGS > AUTO DETECT > AUTOMATIC > DO NOT SET > MANUAL ( DNS1 : 184.108.40.206 , DNS2 : 220.127.116.11 ) > AUTO > DO NOT USE > ENABLE > X > X
2. WIRELESS > ENTER MANUALLY > ( ENTER SSID OR CONTINUE ) > ( CHOOSE SECURITY OR CONTINUE ) > CUSTOM > AUTO > DO NOT SET > MANUAL ( DNS1 : 18.104.22.168 , DNS2 : 22.214.171.124 ) > AUTO > DO NOT USE > ENABLE > X > X
That works for me. Hope I helped