To answer your question. I would give an analogous answer to clarify it better. Apache Spark is also one of the most prominent big data tools in the market. It too uses an In-Memory computation to run tasks quickly but also utilizes clusters computing for better distribution of the workload.
Similar to that R+ Hadoop also works in the same fashion. What R effectively does is define the tasks or Machine Learning Algorithms to use and it translates over the nodes of the Hadoop cluster to better utilize the parallelization aspect.
Since the usual working of the R language is very RAM dependent this makes the processing time of massive data exponentially high. Usually, R solves this problem by allowing multithreading to take place within the cores so it can run the task as optimized as possible. But as you can tell this provides an upper cap on how much data it can handle. What R + Hadoop does is it provides more nodes so essentially translates to more cores it can use to run the following task.
Edit: As requested more elaboration on the last point
Simply put when looking at just R as a language which is optimally built for data analytics, it utilizes multithreading to perform the required analytical tasks more effectively. This multithreading feature lets you run different tasks as different threads simultaneously so thereby trying to achieve as much as parallelization as possible within the limited capabilities of R.
And to answer your question whether R can perform the same tasks as R+Hadoop provided if there are no time constraints
The answer is theoretically Yes practically No!. Intuitively we might think R can handle such tasks by consuming more time. But that isn't always the case. It depends more on the task.
For example, if R were doing a simple frequency Count of the number occurrences in an array of a few million records you can obviously say that even with a few threads it might be slower than R+Hadoop and finish the task after a certain amount of time. But if the task were to say something like Eigen Value Decomposition of a Matrix, such a transformation requires a lot of calculation at the element level of a matrix. Assuming that the size of that matrix is a few million x few million, this type of calculation will surely cause a memory error using just any normal application as it will bottleneck all the data it has to hold temporarily into the RAM. This bottlenecking problem might cause the whole system to fail at times but the most simple case is that of a memory error.
Using R+Hadoop solves such problems as it lets you distribute the same strategy over multiple computers with their corresponding cores. Hence it lets you handle loads much more easily than running it on the same application, this manner you can also prevent the errors which I mentioned above. This is the power of distributed computing.