macOS Ventura 13.6.3 does not open Accessibility on System Preference

If it is not working, I don't think you can regain access without rebooting.

If the problem persists after rebooting, try rebooting in Safe Mode. Does it work there? Reboot normally again. Does it continue to work?

If the problem still persists: Please run Etrecheck and post its full report here. Use the "additional text" button and paste the report into the text box.

    Antivirus software: Apple and nProtect

Uninstall any third party "antivirus" or "cleaners". They do not protect you, cause performance and other issues.

For one, System Preferences was used in macOS 12 Monterey

It was renamed System Setting in macOS 13 Ventura and above

Two, what is presented in above image is " Appearances " and not Accessibilities

Accessibilities would like like in image below

Follow the advise from @ Luis Sequeira1

Download the Application Etrecheck directly from the Developer.

The Application is Not a " Silver Bullet " but only a tool to examine the Hardware / Software used on this computer 

This is a Diagnostic Tool that makes no changes to the computer.

It makes a coherent and readable inventory of both the Hardware and Software used on the computer 

The application is free or paid for added features. 

The Report will Not Reveal Any Personal Information. 

Post back the Full Report - copy and paste - >>>> using the Additional Text Icon  <<<<

Using EtreCheck

prko wrote:

This often happens.... This is my problem. I cannot reach Apple developers.

None of us can reach Apple developers.

If this is happening often to you, you should know that it is very uncommon (actually, your report is the first of its kind here, AFAICT). When uncommon errors occur, it is usually due to something specific to that machine, rather than an OS bug.

That is why I suggested you post the Etrecheck report: by showing what third party software modifications are installed, we often get a clue as to what might be causing the problem in the first place.

A very very fast review of the report

Part 1 of 2

Size: 4.00 TB

Free: 103.49 GB

Available: 115.51 GB

This computer has reached the Critical Point where it will eventually Cease to Boot Up even in Safe Mode

Most anything else this time is secondary

Part 2 of 2

The final word from Apple on Managing the " System Data “ Category

System Data: Contains files that don’t fall into the categories listed here. This category primarily includes files and data used by the system, such as log files, caches, VM files, and other runtime system resources. Also included are temporary files, fonts, app support files, and plug-ins.

You can't manage the contents of this category. The contents are managed by macOS, and the category varies in size depending on the current state of your Mac.

There are areas of the Operating System which we users have control over and that is the User Account Folder ( Home Folder )

All other areas are not accessible 

It is generally a good computer practice to alway keep at least 20% to 25% of the Total Drive Capacity’s as Empty Space.

Allowing the computer to drop below these guidelines may eventually, cause unintended consequences including what is happening Now on this computer

There is Purgeable Space and there is Empty Space.

Purgeable Space which is Controlled by the Operating System. When the Operating Systems decides the computer needs additional Empty Space, it will move a portion of the Purgeable to Empty space

AFAIK - there is no User Actions to hasten this transition from Purgeable to Empty Space

It can day or longer before this will occur.

The links below will assist in identifying what is taking up space on the Internal Drive and provide possible ways to remove data that is under the direct control of the User ( Home Folder ) . 

Rebuild the Spotlight index on your Mac

What is “Other” storage on a Mac, and how can I clean it out?

Free up storage space on your Mac

GrandPerspective 

How to delete Time Machine snapshots on your Mac.  

Often caused if the Time Machine Drive has not been attached  to the computer and TM Backup is set to run on a Schedule. 

TM Backup will make Snap Shot on the Internal Drive awaiting the TM Backup Drive to be attached. 

Only then will the Snaps Shots be transferred to the External Drive and deleted the Internal Drive.

View APFS snapshots in Disk Utility on Mac

See used and available storage space on your Mac

Locate backups of your iPhone, iPad, and iPod touch

Is there a Suite of Adobe Applications  used on this computer ?

They may create some very large cache files that can be removed. Though, the Adobe cache files will be recreated as the Applications needs them.

https://helpx.adobe.com/ca/premiere-pro/kb/clear-cache.html

I reiterate the first point by PRP_53: make some free space as soon as possible. A 4TB drive with only about 100GB free not only hampers performance severely, it can soon fail to boot and incur data loss.

😎 good catch