Is it logistically possible to work on two PhD research projects at the same time? Say someone wants to enroll in two unrelated disciplines: e.g. a Ph.D. in, say, Arabic Literature, and Ph.D. in, say, in Math Education. 

I am not going into the capability of the candidate, or whether it is reasonable to seek multiple PhDs. 

Is it logistically possible to work on two research projects at the same time?

What difficulties/barriers may come?
Well, anything is possible that isn't impossible. But the obstacles would make it extremely unlikely if they are quite different fields. It would also probably result in work that is less than superior, which the same sort of person might achieve had they focused on one field (at a time). Some obstacles:

A single university would have a hard time accepting it, so you need to deal with that. Two different universities permitting you to pursue degrees simultaneously? Maybe you need to hide each from the other. This could lead to problems if either becomes aware. 

A given advisor would be likely to oppose it, thinking that the person isn't giving enough to studies in that field. Do you need, again, to hide what you are doing? 

You may need to pay for at least one of the programs as getting TAs or whatever in two fields is both hard and would require too much time in itself. (Varies by country, of course)

You need to do what ever coursework is necessary in both programs to be able to pass two very different sets of comprehensive exams (many/most places). 

You need, of course, to do the required research in some narrow area in two non-synergistically aligned fields. 

And, if your "friend" just wants to do it as a challenge, sure, why not. Spend six or seven years (more?) chasing a crazy goal just for the hell of it. And possibly fail at both. Seven years older, deeper in debt. 

Now, having succeeded, you need to find a job, probably abandoning everything you did in one of the fields. This last is what actually makes it sort of foolish. It would be a difficult (impossible?) quest to get multiple appointments in a single university and many universities wouldn't permit you to hold a regular appointment at another. 

Some people do get multiple terminal degrees, though usually more closely aligned. Some people find themselves dissatisfied in the field they first qualify in and move to something else. But even these cases are pretty rare. 

Fun as a thought experiment, though.