Those "Out of Memory" errors will be in reference to virtual memory, not physical memory. It is an indication of hitting something called the system commit limit, which is calculated based on the amount of physical memory of the system plus the size of the paging file(s). When processes allocate certain types of virtual memory, the allocation is charged against the commit limit. Windows is making you a commitment it will be able to back that data by physical memory or store it on disk in a paging file. Once you have reached the commit limit, Windows can no longer give you a guarantee it will have some form of medium to store that data and will therefore fail the allocation.
Other than the fact that the amount of physical memory installed in the machine and the size of the paging file make up a system wide limit, there is no correlation between physical and virtual memory. A process merely allocating committed virtual memory will only have an impact on the system committed virtual memory requirements of the system. It is only when the process touches those pages of memory does it demand it to be backed by physical memory.
The role of the paging file, despite a lot of people's misunderstandings on the subject, isn't about satisfying the system when it runs out of physical memory. The purpose of the paging file is to enable the system to page things out which are not otherwise backed by files on disk. When for instance executable files or DLL's need to be removed from physical memory, they don't need to be written to disk beforehand, they can simply be ejected. If the system needs those pieces of code again, they will simply be retrieved from the original file location. This is opposed to pages of memory that are private to a process, which if ejected from memory will be written to the paging file, assuming you have one, of course.
It's also worth saying the paging file isn't some kind of mystical creature which is an absolute necessity in order for Windows to function correctly. The system isn't going to turn radioactive as soon as the paging file is disabled. If there is no paging file, the private pages of memory which are sitting on the modified page list waiting to be written out to disk will need to stay resident in physical memory (on the modified page list) because there isn't anywhere for them to be paged out too. While this will result in less physical memory being available for other purposes, it doesn't cause any issue in itself.
The actual issue which arises from disabling the paging file is related to the system commit limit. If there is no paging file, the commit limit can only ever be the value equal to the amount of physical memory of the system. If your workload requires more system committed virtual memory than the amount of RAM you have, you’re going to run into those "Your system is low on memory" errors, which is in a best case scenario.
Here is some more information on the subject if anyone is interested:
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