Can an audio recording tell whether it is made in a large or small room?
There could be a number of approaches for this task, depending on the number of channels (microphones) in your input and structure of available data.

Given a labeled data set from a previously known set of rooms, you can train a neural network (NN) to classify in which one of them a new sample was recorded. 

A second approach would be to estimate the parameters of the room impulse response (RIR). Certain methods could be under the limitation of a single microphone. A microphone array would supply more information like in this paper. Depending on the shape of the array, different methods can be utilized.

Also, do you have any other prior knowledge on the room? For example, can you get a clean (without speech) audio segment? Do you know where is the microphone and sound sources are located on the room? These priors can point you to methods that utilize this information and improve the estimation.