My iPhone is saying "No SIM"

If you see No Service or Searching on your iPhone or iPad

1. Check your coverage area. Make sure that you're in an area with cellular network coverage. ... 
2. Restart your iPhone or iPad. Restart your device. ... 
3. Check for a Carrier Settings update. ... 
4. Take out the SIM card. ... 
5. Reset your Network Settings. ... 
6. Update your iPhone or iPad. ... 
7. Contact your carrier. ... 
8. Get more help.

If you see No Service or Searching on your iPhone or iPad ...

Remove your SIM card from the SIM card tray and then put the SIM card back. ... Make sure that the SIM tray closes completely and isn't loose. If you use a SIM tray from a different model iPhone, iPad, or from another phone manufacturer, it might not fit properly.

If you see 'invalid SIM' or 'no SIM' on your iPhone or iPad ...

Learn what to do if you see a SIM not supported alert.

Why can't you take the sim out? It would seem the sim card lost connection with the iPhone. Make sure you are pressing the small hole on the side of the iPhone with enough pressure on the sim eject tool or paper clip to pop the tray out.

If the tray is not coming out at all with any amount of pressure, then set up a Genius Appointment at an Apple Store to have it checked out, the sim may have come dislodged form the tray and may be preventing the tray from popping out.

iPhone Repair - Official Apple Support