Keyless extry systems generally entail a dedicated remote lock/unlock feature - ability to unlock/lock the car without the use of the physical key. The one you are thinking about are the step-up hands-free keyless entry or proximity based keyless entry sometimes called by their marketing names of "Smart Key System", "Intelligent or Advanced Key System", "Keyless Go", or similar. Some cars have them, but not all - the issue there is the radio emissions from many of those RF systems - some at power levels that can interfere with defibrillators, pacemakers, and some other RF devices.
Can you add such a system to a 2005 Corolla - Sure. Will it be expensive - Depends on your definition of expensive and who will be doing the electronic work. The base modules themselves don't cost much more than the standard keyless entry systems, the added ECM circuitry on these passive systems, one the other hand, are quite expensive. Used to be a $350 option on certain model Toyotas.
Some have been able to repurpose their existing keyless extry system with an add on module with some additional wiring. There are several aftermarket options - some use RFID keytags, others use a Bluetooth enabled phone as the "key". Costs vary wildly, anywhere from $200-$300 for the base module to easily into the thousands, depending on features. Sure it can be nice, but generally not as secure as some of the OEM setups - unless they have a code jumping feature or RSA (relay station attack) prevention algorithms. Bluetooth protocol, itself, is not very secure wireless communication and RFID tags can be cloned at a distance.
Thanks for the reply. It's clear now. I am not planning on spending extra money to add the hands-free feature but thank you for listing those options - great for future knowledge. However, it doesn't make sense to me that my car auto locks itself if the remote is not within the "bubble" for a few minutes. If that is the case, isn't it possible make it unlock when I approach the "bubble" as it senses the remote? Is there a way?