How can I raytrace a scene that does not fit into memory? If the scene to be raytraced cannot be stored in memory, then without adding more RAM to the machine it seems unrealistic to render it in a practical time span, due to the need to load different parts of the scene from disk potentially several times per pixel.

Is there any way around this? I'm trying to think of some way of performing a large number of the calculations involving a particular subset of the scene all at once, to reduce the number of times it needs to be loaded into memory. Is there any other way of improving the speed in such a case?
If the scene does not entirely fit into memory, you are entering the field of out-of-core rendering. There are essentially two approaches here: a) Generate your scene on-demand b) Load your scene on-demand

The former approach aligns well with most animation workflows, where models are heavily subdivided using e.g. Catmull-Clark and can become very memory-intensive, but the base meshes themselves easily fit into memory. Pixar have a few papers about this (e.g. Ray Differentials and Multiresolution Geometry Caching
for Distribution Ray Tracing in Complex Scenes), but the gist of it is that models are only subdivided when they are hit by a ray, and only subdivided as much as is reasonable for such a ray (e.g. diffuse interreflection need less accuracy than mirror reflections). The rest is handled by a geometry cache, which keeps the subdivided models in memory and hopefully makes the process efficient by a good eviction strategy.

As long as all your base meshes comfortably fit into memory, you can easily go out-of-core and render meshes at subdivision levels that would never fit into memory. The geometry cache also scales nicely with the amount of memory you have, allowing you to weigh RAM vs. render times. This was also used in Cars I believe.

The second approach is more general and does not rely on heavy use of subdivision. Instead, it relies on the fact that your scene was most likely made by an artist and already comes partitioned into reasonably small objects that fit into memory individually. The idea is then to keep two hierarchies (kD-tree or bounding volume hierarchy): A top-level hierarchy that only stores bounding boxes of the objects in your scene, and a low-level hierarchy that stores the actual geometry. There is one such low-level hierarchy for each object.

In this approach, you ideally already store a bounding box along with each object on disk. As the scene is loaded, you only build the top-level hierarchy initially, meaning you only have to look at the bounding boxes and not the geometry. You then start tracing rays and traverse them through the hierarchy. Whenever a ray hits a leaf node in the top-level hierarchy (i.e. it hits the bounding box of an object), that object is loaded into memory and its low-level hierarchy is built. The ray then continues down into tracing that object. Combined with an object cache that keeps as much of the low-level hierarchy in memory as possible, this can perform reasonably well.

The first benefit of such an approach is that objects that are never hit are never loaded, meaning that it automatically adapts to the visibility in your scene. The second benefit is that if you are tracing many rays, you don't have to load an object immediately as it is hit by a ray; instead, you can hold that ray and wait until enough rays have hit that object, amortizing the load over multiple ray hits.

You can also combine this approach with a ray sorting algorithm such as Sorted Deferred Shading for Production Path Tracing to avoid thrashing due to incoherent rays. The mentioned paper describes the architecture of Disney's Hyperion renderer, used for Big Hero 6 I believe, so it most likely can handle scenes at production scale.