How can you have p < 0.0001 with a sample size of 89 and a control group of 52? From Efficacy of monoterpene perillyl alcohol upon survival rate of patients with recurrent glioblastoma (linked from Does Frankincense cure cancer?):

Patients and methods
It was included 89 adults with recurrent GBM receiving daily intranasal administration of 440 mg POH and 52 matched GBM patients as historical control untreated group only with supportive treatment.
Results
Patients with recurrent primary GBM treated with POH survived significantly longer (log rank test, P < 0.0001) than untreated group.

Is it even possible to have p < 0.0001 for such a small sample & control group? I'm not a statistician, but I thought you'd need tens of thousands of samples at the very least to get that sort of confidence. The Wikipedia article on log-rank test didn't really tell me whether p-values need to be interpreted differently for this kind of test, so I'm curious what to make of it.
 A: This is not intended to compete with AdamO's answer -- I'm going to address some specific issues raised in your question and answer somewhat more generally.

Is it even possible to have p < 0.0001 for such a small sample & control group?

Indeed, tiny p-values are possible with much smaller groups than those; this applies whether one deals with parametric or nonparametric tests, and the log-rank test is no exception.
Unless you've got some experience with these things, intuition about the sample sizes required to achieve low p-values can be surprisingly poor.

I'm not a statistician, but I thought you'd need tens of thousands of samples at the very least to get that sort of confidence. 

A p-value is not really "confidence"; in particular, you must beware any interpretation that would consider it as "confidence that the null is false" or "confidence that the alternative is true". 

The Wikipedia article on log-rank test didn't really tell me whether p-values need to be interpreted differently for this kind of test

No. The p-value here is still just a p-value with the usual interpretation of p-values, but all the usual considerations and caveats for the calculation of p-values to be correct apply.

A comment in respect of the headline of the article that links to the paper:

Does Frankincense cure cancer?

The statistical results presented in the abstract of the paper do not in any sense correspond to "curing cancer" (a bit of a linkbait headline, perhaps, common with reporting on scientific papers). They relate to differences in survival time (or as the paper clearly puts it "delay towards progression").
A: If nearly all the failures occur in one group and not in another, the $p$-value will be nearly 0 regardless of how many individuals are in a group. There are many reasons for this. In particular there is small sample bias. For time-to-event analyses, the power is not driven by the total sample size, but the number of failures. No failures, no power, no matter how many participants in the sample.
One nice connection between survival analysis approaches is the relationship between the Log Rank test and the Cox Model. Just as the score test for a logistic regression model on a binary indicator gives on the same $p$-value as is obtained from a Pearson chi-square test for $2 \times 2$ contingency tables, the Score test for the Cox Proportional Hazards model gives the same $p$-value as is obtained from the Log Rank test. In essence, they are consistent tests, and will give you the same answers "in the long run". 
This is important because by fitting a Cox model, you estimate a hazard ratio which allows you to numerically summarize the difference in survival between both groups. Examining the 95% confidence interval from the Cox model would be a nice way to "visualize" the uncertainty in these estimates rather than a very small p-value, I bet what you see is an extremely wide confidence interval.
So, yes it's possible to obtain these kinds of "crazy" $p$-values in categorical and survival analyses because of small sample bias. The power of the analysis depends on how many failures are in each group. Summarizing the power would be a useful way, also, to better understand the problem. If you were treating this test in a Fisherian sense, you would have to report the power of the test before presenting the $p$-value. This would calibrate our expectations as to how significant we really expect findings leading to a $p$-value of 0.0001 would be.
