The space for voicemails is on the carrier’s voicemail server, so that message means that the voicemail server is full, not your iPhone. Most voicemail servers hold 40 messages.
Go to Phone/Voicemail, and tap on Deleted Voicemails. If there are any there, clear them. Note that deleting voicemails requires a cellular data connection, so if you delete from your iPhone and you don’t have cellular data enabled (not Wi-Fi) the messages will not be removed from the carrier’s server.
The next step would be to log in to the carrier’s voicemail and delete the messages there. Follow your carrier’s instructions, but what usually works is to call your own number or hold the “1” key down on the iPhone dial pad for 2 seconds. Then log in with your voicemail passcode and follow the voice menu instructions to review and delete your messages.